Question: Evaluate the double integral: $\int_{0}^{1} \int_{1}^{2}(1-y) x^{2} d x d y$

Answer: $\frac{7}{6} $


Question: Find the Laplacian of the function $f(x, y, z)=x+y+z$.

Answer: $0 $


Question: Find the constrained maxima and minima of $f(x, y)=2 x+y$ given that $x^{2}+y^{2}=4$.

Answer: $\max .\left(\frac{4}{\sqrt{5}}, \frac{2}{\sqrt{5}}\right) $


Question: Calculate $\int_{C} \mathbf{f} \cdot d \mathbf{r}$ for the given vector field $\mathbf{f}(x, y, z)$ and curve $C$: $\mathbf{f}(x, y, z)=\mathbf{i}-\mathbf{j}+\mathbf{k} ; \quad C: x=3 t, y=2 t, z=t, 0 \leq t \leq 1$

Answer: 2


Question:
Calculate the line integral $\int_{C} f(x, y) d s$ for the given function $f(x, y)$ and curve $C$.
$f(x, y)=x y ; \quad C: x=\cos t, y=\sin t, 0 \leq t \leq \pi / 2$

Answer: $1 / 2$


Question: State the domain and range of the given function: $f(x, y)=\sqrt{x^{2}+y^{2}-4}$

Answer: domain: $\left\{(x, y): x^{2}+y^{2} \geq 4\right\}$, range: $[0, \infty) $


Question: Find the volume $V$ of the solid $S$ bounded by the three coordinate planes, bounded above by the plane $x+y+z=2$, and bounded below by the plane $z=x+y$.

Answer: $\frac{1}{3}$


Question: Is there a potential $F(x, y)$ for $\mathbf{f}(x, y)=\left(y^{2}+3 x^{2}\right) \mathbf{i}+2 x y \mathbf{j}$ ? If so, find one.

Answer: Yes. $F(x, y)=x y^{2}+x^{3}$


Question:
Find the center of mass of the region $R$ with the given density function $\delta(x, y)$:
$R=\left\{(x, y): y \geq 0, x^{2}+y^{2} \leq a^{2}\right\}, \delta(x, y)=1$

Answer: $\left(0, \frac{4 a}{3 \pi}\right) \quad$


Question: Evaluate the given double integral: $\int_{0}^{\infty} \int_{0}^{\infty} x y e^{-\left(x^{2}+y^{2}\right)} d x d y$

Answer: $\frac{1}{4} $


Question: Evaluate the limit: $\lim _{(x, y) \rightarrow(1,-1)} \frac{x^{2}-2 x y+y^{2}}{x-y}$

Answer: $2 $


Question: Evaluate the limit: $\lim _{(x, y) \rightarrow(0,0)} \cos (x y)$

Answer: $1 $


Question: Calculate $\int_{C} f(x, y, z) d s$ for the given function $f(x, y, z)$ and curve $C$: $f(x, y, z)=z^{2} ; \quad C: x=t \sin t, y=t \cos t, z=\frac{2 \sqrt{2}}{3} t^{3 / 2}, 0 \leq t \leq 1$

Answer: $2 / 5 $


Question: Let $f(x, y, z)=e^{-x^{2}-y^{2}-z^{2}}$ in Cartesian coordinates. Find the Laplacian of the function in spherical coordinates.

Answer: $\left(4 \rho^{2}-6\right) e^{-\rho^{2}} $


Question: Find the angle $\theta$ between the vectors $\mathbf{v}=(2,1,4)$ and $\mathbf{w}=(1,-2,0)$.

Answer: $90^{\circ} $


Question: Find the constrained maxima and minima of $f(x, y, z)=x+y^{2}+2 z$ given that $4 x^{2}+9 y^{2}-36 z^{2}=36$.

Answer: $\frac{8 a b c}{3 \sqrt{3}}$


Question: Find the volume inside the elliptic cylinder $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}} \leq 1$ for $0 \leq z \leq 2$.

Answer: $2 \pi a b$


Question:
Calculate the area of the parallelogram $P Q R S$:
$P=(2,1,3), Q=(1,4,5), R=(2,5,3), S=(3,2,1)$

Answer: 4 \sqrt{5}


Question: Evaluate the given triple integral: $\int_{1}^{e} \int_{0}^{y} \int_{0}^{1 / y} x^{2} z d x d z d y$

Answer: $\frac{1}{6} $


Question: State whether or not the vector field $\mathbf{f}(x, y, z)$ has a potential in $\mathbb{R}^{3}$ (you do not need to find the potential itself): $\mathbf{f}(x, y, z)=a \mathbf{i}+b \mathbf{j}+c \mathbf{k}(a, b, c$ constant $)$

Answer: Yes


Question:
Calculate the line integral $\int_{C} \mathbf{f} \cdot d \mathbf{r}$ for the given vector field $\mathbf{f}(x, y)$ and curve $C$.
$\mathbf{f}(x, y)=\left(x^{2}-y\right) \mathbf{i}+\left(x-y^{2}\right) \mathbf{j} ; \quad C: x=\cos t, y=\sin t, 0 \leq t \leq 2 \pi$

Answer: $2 \pi$


Question:
Calculate $\mathbf{u} \cdot(\mathbf{v} \times \mathbf{w})$ and $\mathbf{u} \times(\mathbf{v} \times \mathbf{w})$:
$\mathbf{u}=(1,1,1), \mathbf{v}=(3,0,2), \mathbf{w}=(2,2,2)$

Answer: 0 and $(8,-10,2)$


Question: Let $f(x, y, z)=\left(x^{2}+y^{2}+z^{2}\right)^{3 / 2}$ in Cartesian coordinates. Find the Laplacian of $f$ in spherical coordinates.

Answer: $12 \rho $


Question: Determine if the given equation describes a sphere. If so, find its radius and center: $2 x^{2}+2 y^{2}+2 z^{2}+4 x+4 y+4 z-44=0$

Answer: radius: 5, center: $(-1,-1,-1)$


Question: Evaluate the limit: $\lim _{(x, y) \rightarrow(0,0)} \frac{x^{2}-y^{2}}{x^{2}+y^{2}}$

Answer: does not exist


Question: Evaluate the limit: $\lim _{(x, y) \rightarrow(0,0)} \frac{y^{4} \sin (x y)}{x^{2}+y^{2}}$

Answer: $0 $


Question: Find the equation of the tangent plane to the surface $z=f(x, y)$ at the point $P$: $f(x, y)=x+2 y$, $P=(2,1,4)$.

Answer: $x+2 y=z $


Question: Calculate $\int_{C} f(x, y, z) d s$ for the given function $f(x, y, z)$ and curve $C$: $f(x, y, z)=z ; \quad C: x=\cos t, y=\sin t, z=t, 0 \leq t \leq 2 \pi$

Answer: $2 \sqrt{2} \pi^{2} $


Question: Find the angle $\theta$ between the vectors $\mathbf{v}=(5,1,-2)$ and $\mathbf{w}=(4,-4,3)$.

Answer: $73.4^{\circ} $


Question:
Calculate $(\mathbf{u} \times \mathbf{v}) \cdot(\mathbf{w} \times \mathbf{z})$:
$\mathbf{u}=(1,1,1), \mathbf{v}=(3,0,2), \mathbf{w}=(2,2,2), \mathbf{z}=(2,1,4)$

Answer: $14 $


Question:
Evaluate the integral

$$
\int_{-\infty}^{\infty} e^{-x^{2}} d x
$$

using anything you have learned so far.

Answer: $\sqrt{\pi}$


Question: How many seven-element subsets are there in a set of nine elements?

Answer: $\left(\begin{array}{l}9 \\ 7\end{array}\right)=36$


Question: A die is loaded so that the probability of a face coming up is proportional to the number on that face. The die is rolled with outcome $X$. Find $V(X)$ and $D(X)$.

Answer: $V(X)=\frac{20}{9}, \quad D(X)=\frac{2 \sqrt{5}}{3}$.


Question: Modify the program AllPermutations to count the number of permutations of $n$ objects that have exactly $j$ fixed points for $j=0,1,2, \ldots, n$. Run your program for $n=2$ to 6 . Make a conjecture for the relation between the number that have 0 fixed points and the number that have exactly 1 fixed point. A proof of the correct conjecture can be found in Wilf. ${ }^{12}$

Answer: They are the same.


Question: A coin is tossed three times. Let $X$ be the number of heads that turn up. Find $V(X)$ and $D(X)$.

Answer: $V(X)=\frac{3}{4}, \quad D(X)=\frac{\sqrt{3}}{2}$.


Question: The lifetime, measure in hours, of the ACME super light bulb is a random variable $T$ with density function $f_{T}(t)=\lambda^{2} t e^{-\lambda t}$, where $\lambda=.05$. What is the expected lifetime of this light bulb? What is its variance?

Answer: $\mu=40, \sigma^{2}=800$


Question: In a second version of roulette in Las Vegas, a player bets on red or black. Half of the numbers from 1 to 36 are red, and half are black. If a player bets a dollar on black, and if the ball stops on a black number, he gets his dollar back and another dollar. If the ball stops on a red number or on 0 or 00 he loses his dollar. Find the expected winnings for this bet.

Answer: $-1 / 19$


Question: Use the program BinomialProbabilities to find the probability that, in 100 tosses of a fair coin, the number of heads that turns up lies between 35 and 65 , between 40 and 60 , and between 45 and 55 .

Answer: .998,.965,.729


Question: Four people are to be arranged in a row to have their picture taken. In how many ways can this be done?

Answer: 24


Question: Assume that the probability that there is a significant accident in a nuclear power plant during one year's time is .001. If a country has 100 nuclear plants, estimate the probability that there is at least one such accident during a given year.

Answer: $m=100 \times(.001)=.1$. Thus $P$ (at least one accident $)=1-e^{-.1}=.0952$.


Question: You have 80 dollars and play the following game. An urn contains two white balls and two black balls. You draw the balls out one at a time without replacement until all the balls are gone. On each draw, you bet half of your present fortune that you will draw a white ball. What is your expected final fortune?

Answer: 45


Question: A card is drawn at random from a deck consisting of cards numbered 2 through 10. A player wins 1 dollar if the number on the card is odd and loses 1 dollar if the number if even. What is the expected value of his winnings?

Answer: $-1 / 9$


Question: At the Tunbridge World's Fair, a coin toss game works as follows. Quarters are tossed onto a checkerboard. The management keeps all the quarters, but for each quarter landing entirely within one square of the checkerboard the management pays a dollar. Assume that the edge of each square is twice the diameter of a quarter, and that the outcomes are described by coordinates chosen at random. Is this a fair game?

Answer: Yes.


Question: A computing center has 3 processors that receive $n$ jobs, with the jobs assigned to the processors purely at random so that all of the $3^{n}$ possible assignments are equally likely. Find the probability that exactly one processor has no jobs.

Answer: $\frac{\left(\begin{array}{l}3 \\ 1\end{array}\right) \times\left(2^{n}-2\right)}{3^{n}}$.


Question: In a poker hand, John has a very strong hand and bets 5 dollars. The probability that Mary has a better hand is .04. If Mary had a better hand she would raise with probability .9 , but with a poorer hand she would only raise with probability .1. If Mary raises, what is the probability that she has a better hand than John does?

Answer: .273.


Question: A process moves on the integers 1, 2, 3, 4, and 5. It starts at 1 and, on each successive step, moves to an integer greater than its present position, moving with equal probability to each of the remaining larger integers. State five is an absorbing state. Find the expected number of steps to reach state five.

Answer: 2.08


Question: The probability of a royal flush in a poker hand is $p=1 / 649,740$. How large must $n$ be to render the probability of having no royal flush in $n$ hands smaller than $1 / e$ ?

Answer: 649741


Question: It is desired to find the probability that in a bridge deal each player receives an ace. A student argues as follows. It does not matter where the first ace goes. The second ace must go to one of the other three players and this occurs with probability $3 / 4$. Then the next must go to one of two, an event of probability $1 / 2$, and finally the last ace must go to the player who does not have an ace. This occurs with probability $1 / 4$. The probability that all these events occur is the product $(3 / 4)(1 / 2)(1 / 4)=3 / 32$. Is this argument correct?

Answer: No.


Question: One of the first conditional probability paradoxes was provided by Bertrand. ${ }^{23}$ It is called the Box Paradox. A cabinet has three drawers. In the first drawer there are two gold balls, in the second drawer there are two silver balls, and in the third drawer there is one silver and one gold ball. A drawer is picked at random and a ball chosen at random from the two balls in the drawer. Given that a gold ball was drawn, what is the probability that the drawer with the two gold balls was chosen?

Answer: $2 / 3$


Question: A lady wishes to color her fingernails on one hand using at most two of the colors red, yellow, and blue. How many ways can she do this?

Answer: $3\left(2^{5}\right)-3=93$ (we subtract 3 because the three pure colors are each counted twice).


Question: Let $X$ be a random variable with $E(X)=0$ and $V(X)=1$. What integer value $k$ will assure us that $P(|X| \geq k) \leq .01$ ?

Answer: $k=10$


Question: A number is chosen at random from the set $S=\{-1,0,1\}$. Let $X$ be the number chosen. Find the expected value, variance, and standard deviation of $X$.

Answer: $E(X)=0, V(X)=\frac{2}{3}, \quad \sigma=D(X)=\sqrt{\frac{2}{3}}$.


Question: Let $X$ be a random variable having a normal density and consider the random variable $Y=e^{X}$. Then $Y$ has a $\log$ normal density. Find this density of $Y$.

Answer: $F_{Y}(y)=\frac{1}{\sqrt{2 \pi y}} e^{-\frac{\log ^{2}(y)}{2}}$, for $y>0$.


Question: A student must choose exactly two out of three electives: art, French, and mathematics. He chooses art with probability $5 / 8$, French with probability $5 / 8$, and art and French together with probability $1 / 4$. What is the probability that he chooses mathematics? What is the probability that he chooses either art or French?

Answer: $3 / 4,1$


Question: A fair coin is tossed 100 times. The expected number of heads is 50, and the standard deviation for the number of heads is $(100 \cdot 1 / 2 \cdot 1 / 2)^{1 / 2}=5$. What does Chebyshev's Inequality tell you about the probability that the number of heads that turn up deviates from the expected number 50 by three or more standard deviations (i.e., by at least 15 )?

Answer: $1 / 9$


Question:
Find integers $n$ and $r$ such that the following equation is true:
$$
\left(\begin{array}{c}
13 \\
5
\end{array}\right)+2\left(\begin{array}{c}
13 \\
6
\end{array}\right)+\left(\begin{array}{c}
13 \\
7
\end{array}\right)=\left(\begin{array}{l}
n \\
r
\end{array}\right)
$$

Answer: $n=15, r=7$


Question: Two cards are drawn from a bridge deck. What is the probability that the second card drawn is red?

Answer: $1 / 2$


Question: It is raining in the Land of Oz. Determine a tree and a tree measure for the next three days' weather. Find $\mathbf{w}^{(1)}, \mathbf{w}^{(2)}$, and $\mathbf{w}^{(3)}$ and compare with the results obtained from $\mathbf{P}, \mathbf{P}^{2}$, and $\mathbf{P}^{3}$.

Answer:
$\mathbf{w}(1)=(.5, .25, .25)$

$\mathbf{w}(2)=(.4375, .1875, .375)$

$\mathbf{w}(3)=(.40625, .203125, .390625)$


Question:
Let $X_{1}$ and $X_{2}$ be independent random variables with common distribution
$$
p_{X}=\left(\begin{array}{ccc}
0 & 1 & 2 \\
1 / 8 & 3 / 8 & 1 / 2
\end{array}\right) .
$$
Find the distribution of the sum $X_{1}+X_{2}$.

Answer: $\quad\left(\begin{array}{ccccc}0 & 1 & 2 & 3 & 4 \\ \frac{1}{64} & \frac{3}{32} & \frac{17}{64} & \frac{3}{8} & \frac{1}{4}\end{array}\right)$


Question: In a class there are 20 students: 3 are 5' 6”, 5 are 5'8”, 4 are 5'10", 4 are 6 ', and 4 are 6' 2". A student is chosen at random. What is the student's expected height?

Answer: $5^{\prime} 10.1^{\prime \prime}$


Question: The Poisson distribution with parameter $\lambda=.3$ has been assigned for the outcome of an experiment. Let $X$ be the outcome function. Find $P(X=0)$, $P(X=1)$, and $P(X>1)$.

Answer: .7408,.2222, .0370


Question: The Yankees are playing the Dodgers in a world series. The Yankees win each game with probability .6. What is the probability that the Yankees win the series? (The series is won by the first team to win four games.)

Answer: .710.


Question: In a digital computer, a bit is one of the integers $\{0,1\}$, and a word is any string of 32 bits. How many different words are possible?

Answer: $2^{32}$


Question: Five people get on an elevator that stops at five floors. Assuming that each has an equal probability of going to any one floor, find the probability that they all get off at different floors.

Answer: $\frac{5 !}{5^{5}}$.


Question: A restaurant offers apple and blueberry pies and stocks an equal number of each kind of pie. Each day ten customers request pie. They choose, with equal probabilities, one of the two kinds of pie. How many pieces of each kind of pie should the owner provide so that the probability is about .95 that each customer gets the pie of his or her own choice?

Answer: Eight pieces of each kind of pie.


Question: A baseball player, Smith, has a batting average of .300 and in a typical game comes to bat three times. Assume that Smith's hits in a game can be considered to be a Bernoulli trials process with probability .3 for success. Find the probability that Smith gets $0,1,2$, and 3 hits.

Answer: .343,.441, .189, .027.


Question: Each of the four engines on an airplane functions correctly on a given flight with probability .99 , and the engines function independently of each other. Assume that the plane can make a safe landing if at least two of its engines are functioning correctly. What is the probability that the engines will allow for a safe landing?

Answer: .999996.


Question: In a horse race, the odds that Romance will win are listed as $2: 3$ and that Downhill will win are $1: 2$. What odds should be given for the event that either Romance or Downhill wins?

Answer: $11: 4$


Question: There are three different routes connecting city A to city B. How many ways can a round trip be made from A to B and back? How many ways if it is desired to take a different route on the way back?

Answer: 9,6 .


Question: Suppose you start with eight pennies and flip one fair coin. If the coin comes up heads, you get to keep all your pennies; if the coin comes up tails, you have to give half of them back. Let $X$ be the total number of pennies you have at the end. Compute $E(X)$.

Answer: $E(X)=6$

The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.


Question: Let $Z \sim$ Negative-Binomial $(3,1 / 4)$. Compute $P(Z \leq 2)$.

Answer: 53/512

The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.


Question: Suppose we repeat a certain experiment 2000 times and obtain a sample average of -5 and a standard error of 17 . In terms of this, specify an interval that is virtually certain to contain the experiment's (unknown) true mean $\mu$.

Answer: $(-6.1404, -3.8596)$

The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.


Question: Suppose $P([0,1])=1$, but $P([1 / n, 1])=0$ for all $n=1,2,3, \ldots$. What must $P(\{0\})$ be?

Answer: 1

The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.


Question:
Let $X_{1}, X_{2}, \ldots, X_{n+1}$ be i.i.d. with distribution $N(0,1)$. Find a value of $C$ such that
$$
C \frac{X_{1}}{\sqrt{X_{2}^{2}+\cdots+X_{n}^{2}+X_{n+1}^{2}}} \sim t(n) .
$$

Answer: $C=\sqrt{n}$

The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.


Question: Consider flipping two fair coins. Let $X=1$ if the first coin is heads, and $X=0$ if the first coin is tails. Let $Y=1$ if the second coin is heads, and $Y=5$ if the second coin is tails. Let $Z=X Y$. What is the probability function of $Z$ ?

Answer: $p_{Z}(1)=p_{Z}(5)=1 / 4, p_{Z}(0)=1 / 2$, and $p_{Z}(z)=0$ otherwise

The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.


Question: Suppose $S=\{1,2,3,4\}$, and $P(\{1\})=1 / 12$, and $P(\{1,2\})=1 / 6$, and $P(\{1,2,3\})=1 / 3$. Compute $P(\{1\}), P(\{2\}), P(\{3\})$, and $P(\{4\})$.

Answer: $P(\{1\})=1 / 12, P(\{2\})=1 / 12, P(\{3\})=1 / 6, P(\{4\})=2 / 3$

The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.


Question: Suppose that the lifelengths (in thousands of hours) of light bulbs are distributed $\operatorname{Exponential}(\theta)$, where $\theta>0$ is unknown. If we observe $\bar{x}=5.2$ for a sample of 20 light bulbs, record a representative likelihood function. Why is it that we only need to observe the sample average to obtain a representative likelihood?

Answer: $L\left(\theta \mid x_{1}, \ldots, x_{20}\right)=\theta^{20} \exp (-(20 \bar{x}) \theta)$ and $\bar{x}$ is a sufficient statistic.

The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.


Question: Suppose you simultaneously roll two dice $n$ times and record the outcomes. Based on these values, how would you assess the null hypothesis that the outcome on each die is independent of the outcome on the other?

Answer: Then there are 36 possible pairs $(i, j)$ for $i, j=1, \ldots, 6$. Let $f_{i j}$ denote the frequency for $(i, j)$ and compute chi-squared statistic, $X^{2}=\sum_{i=1}^{6} \sum_{j=1}^{6}\left(f_{i j}-\right.$ $\left.f_{i} \cdot f_{\cdot j} / n\right)^{2} /\left(f_{i \cdot} \cdot f_{\cdot j} / n\right)$. Compute the P-value $P\left(\chi^{2}(25)>X^{2}\right)$.

The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.


Question: Suppose a statistical model is given by $\left\{f_{1}, f_{2}\right\}$, where $f_{i}$ is an $N(i, 1)$ distribution. Compute the likelihood ratio $L(1 \mid 0) / L(2 \mid 0)$ and explain how you interpret this number.

Answer: $L(1 \mid 0) / L(2 \mid 0)=4.4817$, the distribution $f_{1}$ is 4.4817 times more likely than $f_{2}$.

The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.


Question: Suppose you know that the probability distribution of a variable $X$ is either $P_{1}$ or $P_{2}$. If you observe $X=1$ and $P_{1}(X=1)=0.75$ while $P_{2}(X=1)=0.001$, then what would you guess as the true distribution of $X$ ? Give your reasoning for this conclusion.

Answer: $P_{1}$

The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.


Question: Suppose $P([0,1 / 2])=1 / 3$. Must there be some $n$ such that $P([1 / n, 1 / 2])>$ $1 / 4$ ?

Answer: No

The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.


Question: Suppose you want to test the null hypothesis $H_{0}: \mu=0$ based on a sample of $n$ from an $N(\mu, 1)$ distribution, where $\mu \in\{0,2\}$. How large does $n$ have to be so that the power at $\mu=2$, of the optimal size 0.05 test, is equal to 0.99 ?

Answer: $n \geq 4$

The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.


Question: Let $X \sim \operatorname{Geometric}(\theta)$ and $Y \sim \operatorname{Poisson}(\lambda)$. Compute $E(8 X-Y+12)$.

Answer: $E(8 X-Y+12)=8((1-p) / p)-\lambda+12$

The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.


Question: Suppose we are told only that $P(X>x)=1 / x^{2}$ for $x \geq 1$, and $P(X>x)=1$ for $x<1$, but we are not told if $X$ is discrete or continuous or neither. Compute $E(X)$.

Answer: $E(X)=2$

The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.


Question: Let $X \sim \operatorname{Bernoulli}(1 / 3)$, and let $Y=4 X-2$. Compute the joint $\operatorname{cdf} F_{X, Y}$.

Answer:
$$
F_{X, Y}(x, y)= \begin{cases}0 & \min [x,(y+2) / 4]<0 \\ 1 / 3 & 0 \leq \min [x,(y+2) / 4]<1 \\ 1 & \min [x,(y+2) / 4] \geq 1\end{cases}
$$

The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.


Question: Let $X$ and $Y$ be independent, with $X \sim \operatorname{Bernoulli}(1 / 2)$ and $Y \sim \operatorname{Bernoulli}(1 / 3)$. Let $Z=X+Y$ and $W=X-Y$. Compute $\operatorname{Cov}(Z, W)$ and $\operatorname{Corr}(Z, W)$.

Answer: $\operatorname{Cov}(Z, W)=0, \operatorname{Corr}(Z, W)=0$

The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.


Question: Determine Jeffreys' prior for the Bernoulli $(\theta)$ model and determine the posterior distribution of $\theta$ based on this prior.

Answer: Jeffreys' prior is $\sqrt{n} \theta^{-1 / 2}(1-\theta)^{-1 / 2}$. The posterior distribution of $\theta$ is $\operatorname{Beta}(n \bar{x}$ $+1 / 2, n(1-\bar{x})+1 / 2)$.

The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.


Question: Suppose that a statistical model is given by the family of $N\left(\mu, \sigma_{0}^{2}\right)$ distributions where $\theta=\mu \in R^{1}$ is unknown, while $\sigma_{0}^{2}$ is known. If our interest is in making inferences about the first quartile of the true distribution, then determine $\psi(\mu)$.

Answer: $\psi(\mu)=\mu+\sigma_{0} z_{0.25}$, where $z_{0.25}$ satisfies $\Phi\left(z_{0.25}\right)=0.25$

The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.


Question: Suppose $P(W>w)=e^{-5 w}$ for $w \geq 0$ and $P(W>w)=1$ for $w<0$. Compute $E(W)$.

Answer: $E(W)=1 / 5$

The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.


Question: Suppose that an urn contains a proportion $p$ of chips labelled 0 and proportion $1-p$ of chips labelled 1. For a sample of $n=2$, drawn with replacement, determine the distribution of the sample mean.

Answer: If $Z$ is the sample mean, then $P(Z=0)=p^{2}, P(Z=0.5)=2 p(1-p)$, and $P(Z=1)=(1-p)^{2}$.

The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.


Question: Suppose a baseball pitcher throws fastballs $80 \%$ of the time and curveballs $20 \%$ of the time. Suppose a batter hits a home run on $8 \%$ of all fastball pitches, and on $5 \%$ of all curveball pitches. What is the probability that this batter will hit a home run on this pitcher's next pitch?

Answer: 0.074

The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.


Question: Suppose we define a process $\left\{X_{n}\right\}$ as follows. Given $X_{n}$, with probability $3 / 8$ we let $X_{n+1}=X_{n}-4$, while with probability $5 / 8$ we let $X_{n+1}=X_{n}+C$. What value of $C$ will make $\left\{X_{n}\right\}$ be a martingale?

Answer: $C=12 / 5$

The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.


Question: Let $Y \sim \operatorname{Binomial}(10, \theta)$. Compute $P(Y=10)$.

Answer: $\theta^{10}$

The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.


Question: Suppose we have a statistical model $\left\{f_{\theta}: \theta \in[0,1]\right\}$ and we observe $x_{0}$. Is it true that $\int_{0}^{1} L\left(\theta \mid x_{0}\right) d \theta=1$ ? Explain why or why not.

Answer: No

The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.


Question: Let $X \sim N(3,5)$ and $Y \sim N(-7,2)$ be independent. Find values of $C_{1} \neq$ $0, C_{2}, C_{3} \neq 0, C_{4}, C_{5}$ so that $C_{1}\left(X+C_{2}\right)^{2}+C_{3}\left(Y+C_{4}\right)^{2} \sim \chi^{2}\left(C_{5}\right)$.

Answer: $C_{1}=1 / 5, C_{2}=-3, C_{3}=1 / 2, C_{4}=7, C_{5}=2$

The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.


Question: Let $Z \sim$ Poisson(3). Use Markov's inequality to get an upper bound on $P(Z \geq$ $7)$.

Answer: $3 / 7$

The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.


Question: Explain why it is not possible that the function $\theta^{3} \exp \left(-(\theta-5.3)^{2}\right)$ for $\theta \in R^{1}$ is a likelihood function.

Answer: A likelihood function cannot take negative values.

The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.


Question: Let $\{N(t)\}_{t \geq 0}$ be a Poisson process with intensity $a>0$. Compute (with explanation) the conditional probability $P\left(N_{2.6}=2 \mid N_{2.9}=2\right)$.

Answer: $P\left(N_{2.6}=2 \mid N_{2.9}=2\right)=(2.6 / 2.9)^{2}$

The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.


Question: Suppose that $X \sim \operatorname{Gamma}(3,6)$. What value would you record as a prediction of a future value of $X$ ? How would you justify your choice?

Answer: The mode is $1 / 3$.

The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.


Question: Let $Y=3 X+4$. Compute $m_{Y}(s)$ in terms of $m_{X}$.

Answer: $m_{Y}(s)=e^{4 s} m_{X}(3 s)$

The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.


Question:
Determine the empirical distribution function based on the sample given below.
\begin{tabular}{|rrrrr|}
\hline 1.06 & -1.28 & 0.40 & 1.36 & -0.35 \\
-1.42 & 0.44 & -0.58 & -0.24 & -1.34 \\
0.00 & -1.02 & -1.35 & 2.05 & 1.06 \\
0.98 & 0.38 & 2.13 & -0.03 & -1.29 \\
\hline
\end{tabular}

Using the empirical cdf, determine the sample median, the first and third quartiles, and the interquartile range. What is your estimate of $F(2)$ ?

Answer: The sample median is estimated by -0.03 and the estimate of the first quartile is -1.28 , and for the third quartile is 0.98 . Also $\hat{F}(2)=\hat{F}(1.36)=0.90$.

The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.


Question: Suppose basketball teams $A$ and $B$ each have five players and that each member of team A is being "guarded" by a unique member of team B. Suppose it is noticed that each member of team A is taller than the corresponding guard from team B. Does it necessarily follow that the mean height of team $\mathrm{A}$ is larger than the mean height of team B? Why or why not?

Answer: Yes

The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.


Question: Suppose four fair coins are flipped, and let $Y$ be the number of pairs of coins which land the same way (i.e., the number of pairs that are either both heads or both tails). Compute the exact distribution of $Y$.

Answer: $p_{Y}(y)=1 / 2$ for $y=1$, 2; otherwise, $p_{Y}(y)=0$

The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.


Question: Suppose $X$ has density $f$ and $f(x)>f(y)$ whenever $0<x<1<y<2$. Does it follow that $P(0<X<1)>P(1<X<2)$ ? Explain.

Answer: Yes

The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.


Question: Suppose darts are randomly thrown at a wall. Let $X$ be the distance (in centimeters) from the left edge of the dart's point to the left end of the wall, and let $Y$ be the distance from the right edge of the dart's point to the left end of the wall. Assume the dart's point is 0.1 centimeters thick, and that $E(X)=214$. Compute $E(Y)$.

Answer: $E(Y)=214.1$

The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.


Question: Suppose $X$ has density function $f(x)=3 / 20\left(x^{2}+x^{3}\right)$ for $0<x<2$, otherwise $f(x)=0$. Compute each of $E(X), E\left(X^{2}\right)$, and $E\left(X^{3}\right)$, and rank them from largest to smallest.

Answer: Let $\mu_{k}=E\left(X^{k}\right)$, then $\mu_{1}=39 / 25, \mu_{2}=64 / 25, \mu_{3}=152 / 35$.

The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.


Question: Suppose a statistical model comprises all continuous distributions on $R^{1}$. Based on a sample of $n$, determine a UMVU estimator of $P((-1,1))$, where $P$ is the true probability measure. Justify your answer.

Answer: $n^{-1} \sum_{i=1}^{n} I_{(-1,1)}\left(X_{i}\right)$

The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.


Question: A coin was tossed $n=1000$ times, and the proportion of heads observed was 0.51. Do we have evidence to conclude that the coin is unfair?

Answer: $\mathrm{P}$-value $=0.527$

The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.


Question:
Suppose we obtained the following sample from a distribution that we know has its first six moments. Determine an approximate 0.95-confidence interval for $\mu_{3}$.
\begin{tabular}{|rrrrrrrrrr|}
3.27 & -1.24 & 3.97 & 2.25 & 3.47 & -0.09 & 7.45 & 6.20 & 3.74 & 4.12 \\
1.42 & 2.75 & -1.48 & 4.97 & 8.00 & 3.26 & 0.15 & -3.64 & 4.88 & 4.55 \\
\hline
\end{tabular}

Answer: $m_{3} \pm z_{(1+\gamma) / 2} s_{3} / \sqrt{n}=(26.027,151.373)$

The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.


Question:
Suppose we have a quantitative response variable $Y$ and two categorical predictor variables $W$ and $X$, both taking values in $\{0,1\}$. Suppose the conditional distributions of $Y$ are given by
$$
\begin{aligned}
& Y \mid W=0, X=0 \sim N(2,5) \\
& Y \mid W=1, X=0 \sim N(3,5) \\
& Y \mid W=0, X=1 \sim N(4,5) \\
& Y \mid W=1, X=1 \sim N(4,5) .
\end{aligned}
$$
Does $W$ have a relationship with $Y$ ? Does $X$ have a relationship with $Y$ ? Explain your answers.

Answer: $W$ has a relationship with $Y$ and $X$ has a relationship with $Y$.

The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.


Question: Let $M>0$, and suppose $f(x)=c x^{2}$ for $0<x<M$, otherwise $f(x)=0$. For what value of $c$ (depending on $M$ ) is $f$ a density?

Answer: $c=3 / M^{3}$

The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.


Question: Let $\left\{X_{n}\right\}$ be a martingale, with initial value $X_{0}=5$. Suppose we know that $P\left(X_{8}=3\right)+P\left(X_{8}=4\right)+P\left(X_{8}=6\right)=1$, i.e., $X_{8}$ is always either 3 , 4, or 6 . Suppose further that $P\left(X_{8}=3\right)=2 P\left(X_{8}=6\right)$. Compute $P\left(X_{8}=4\right)$.

Answer: $P\left(X_{n}=4\right)=5 / 8$

The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.


Question: Let $X$ have density function $f_{X}(x)=(1 / 2) \sin (x)$ for $0<x<\pi$, otherwise $f_{X}(x)=0$. Let $Y=X^{2}$. Compute the density function $f_{Y}(y)$ for $Y$.

Answer: $f_{Y}(y)=y^{-1 / 2} \sin \left(y^{1 / 2}\right) / 4$ for $0<y<\pi^{2}$ and 0 otherwise

The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.


Question: Let $X \sim \operatorname{Normal}(0,1)$. Let $Y=X^{3}$. Compute the density function $f_{Y}(y)$ for $Y$.

Answer: $f_{Y}(y)=(2 \pi)^{-1 / 2}\left(3|y|^{2 / 3}\right)^{-1} \exp \left(-|y|^{2 / 3} / 2\right)$

The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.


Question: Suppose that $X$ and $Y$ are random variables such that a regression model describes the relationship between $Y$ and $X$. If $E(Y \mid X)=\beta_{1}+\beta_{2} X^{2}$, then discuss whether or not this is a simple linear regression model (perhaps involving a predictor other than $X$ ).

Answer: We can write $E(Y \mid X)=E\left(Y \mid X^{2}\right)$ in this case and $E\left(Y \mid X^{2}\right)=\beta_{1}+\beta_{2} X^{2}$, so this is a simple linear regression model but the predictor is $X^{2}$ not $X$.

The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.


Question:
Let $X \sim N(3,5)$ and $Y \sim N(-7,2)$ be independent. Find values of $C_{1}, C_{2}, C_{3}$, $C_{4}, C_{5}, C_{6}, C_{7}$ so that
$$
\frac{C_{1}\left(X+C_{2}\right)^{C_{3}}}{\left(Y+C_{4}\right)^{C_{5}}} \sim F\left(C_{6}, C_{7}\right) .
$$

Answer: $C_{1}=2 / 5, C_{2}=-3, C_{3}=2, C_{4}=7, C_{5}=2, C_{6}=1, C_{7}=1$

The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.


Question: Describe a Monte Carlo approximation of $\int_{0}^{\infty} e^{-5 x-14 x^{2}} d x$. (Hint: Remember the Exponential(5) distribution.)

Answer: This integral equals $(1 / 5) E\left(e^{-14 Z^{2}}\right)$, where $Z \sim$ Exponential(5).

The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.


Question: Describe a Monte Carlo approximation of $\int_{-\infty}^{\infty} \cos ^{2}(x) e^{-x^{2} / 2} d x$.

Answer: The integral equals $\sqrt{2 \pi} E\left(\cos ^{2}(Z)\right)$, where $Z \sim N(0,1)$.

The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.


Question: Suppose we deal five cards from an ordinary 52-card deck. What is the conditional probability that the hand contains all four aces, given that the hand contains at least four aces?

Answer: 1

The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.


Question: Suppose an urn contains 1000 balls - one of these is black, and the other 999 are white. Suppose that 100 balls are randomly drawn from the urn with replacement. Use the appropriate Poisson distribution to approximate the probability that five black balls are observed.

Answer: $P(X=5) \approx\left((100 / 1000)^{5} / 5\right.$ ! $) \exp \{-100 / 1000\}$

The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.


Question: If $\left(x_{1}, \ldots, x_{n}\right)$ is a sample from a $\operatorname{Gamma}\left(\alpha_{0}, \theta\right)$ distribution, where $\alpha_{0}>0$ and $\theta \in(0, \infty)$ is unknown, then determine the MLE of $\theta$.

Answer: $\hat{\theta}=\alpha_{0} / \bar{x}$

The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.


Question: Suppose men's heights (in centimeters) follow the distribution $N\left(174,20^{2}\right)$, while those of women follow the distribution $N\left(160,15^{2}\right)$. Compute the mean total height of a man-woman married couple.

Answer: 334

The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.


Question: Suppose $S=\{1,2,3\}$, and $P(\{1\})=P(\{2\})+1 / 6$, and $P(\{3\})=2 P(\{2\})$. Compute $P(\{1\}), P(\{2\})$, and $P(\{3\})$.

Answer: $P(\{2\})=5 / 24, P(\{1\})=3 / 8, P(\{3\})=5 / 12$

The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.


Question: Let $Y \sim$ Poisson $(\lambda)$. Compute $E\left(Y^{3}\right)$, the third moment of $Y$.

Answer: $m_{Y}^{\prime \prime \prime}(s)=e^{\lambda\left(e^{s}-1\right)} e^{s} \lambda\left(1+3 e^{s} \lambda+e^{2 s} \lambda^{2}\right), E\left(Y^{3}\right)=m_{Y}^{\prime \prime \prime}(0)=\lambda\left(1+3 \lambda+\lambda^{2}\right)$

The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.


Question: Suppose that a chi-squared test is carried out, based on a random sample of $n$ from a population, to assess whether or not two categorical variables $X$ and $Y$ are independent. Suppose the P-value equals 0.001 and the investigator concludes that there is evidence against independence. Discuss how you would check to see if the deviation from independence was of practical significance.

Answer: We look at the differences $\left|f_{i j}-f i \cdot f_{\cdot j} / n\right|$ to see how big these are.

The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.


Question: Let $X \sim \operatorname{Exponential}(\lambda)$. Let $Y=X^{3}$. Compute the density $f_{Y}$ of $Y$.

Answer: $f_{Y}(y)$ equals $(\lambda / 3) y^{-2 / 3} e^{-\lambda y^{1 / 3}}$ for $y>0$ and otherwise equals 0 .

The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.


Question: Suppose that a symmetrical die is tossed $n=20$ independent times. Work out the exact sampling distribution of the maximum of this sample.

Answer: For $1 \leq j \leq 6, P(\max =j)=(j / 6)^{20}-((j-1) / 6)^{20}$.

The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.


Question: Suppose that a statistical model is given by the family of $N\left(\mu, \sigma^{2}\right)$ distributions where $\theta=\left(\mu, \sigma^{2}\right) \in R^{1} \times R^{+}$is unknown. If our interest is in making inferences about the distribution function evaluated at 3 , then determine $\psi\left(\mu, \sigma^{2}\right)$.

Answer: $\psi\left(\mu, \sigma^{2}\right)=\Phi((3-\mu) / \sigma)$

The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.


Question: Suppose we define a process $\left\{X_{n}\right\}$ as follows. Given $X_{n}$, with probability $p$ we let $X_{n+1}=2 X_{n}$, while with probability $1-p$ we let $X_{n+1}=X_{n} / 2$. What value of $p$ will make $\left\{X_{n}\right\}$ be a martingale?

Answer: $p=1 / 3$

The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.


Question: Suppose we keep dealing cards from an ordinary 52-card deck until the first jack appears. What is the probability that at least 10 cards go by before the first jack?

Answer: $\left(\begin{array}{c}48 \\ 10\end{array}\right) /\left(\begin{array}{c}52 \\ 10\end{array}\right)=246 / 595=0.4134$

The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.


Question: Let $X_{t}=10-1.5 t+4 B_{t}$. Compute $E\left(X_{3} X_{5}\right)$.

Answer: $E\left(X_{3} X_{5}\right)=61.75$

The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.


Question: Suppose we flip 100 fair independent coins. What is the probability that at least three of them are heads? (Hint: You may wish to use (1.3.1).)

Answer: $1-5051 / 2^{100}$

The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.


Question: Verify that the third moment of an $N\left(\mu, \sigma^{2}\right)$ distribution is given by $\mu_{3}=$ $\mu^{3}+3 \mu \sigma^{2}$. Because the normal distribution is specified by its first two moments, any characteristic of the normal distribution can be estimated by simply plugging in the MLE estimates of $\mu$ and $\sigma^{2}$. Compare the method of moments estimator of $\mu_{3}$ with this plug-in MLE estimator, i.e., determine whether they are the same or not.

Answer: From the mgf, $m_{X}^{\prime \prime \prime}(0)=3 \sigma^{2} \mu+\mu^{3}$. The plug-in estimator is $\hat{\mu}_{3}=3\left(m_{2}-m_{1}^{2}\right) \times$ $m_{1}+m_{1}^{3}$, while the method of moments estimator of $\mu_{3}$ is $m_{3}=\frac{1}{n} \sum x_{i}^{3}$.

The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.


Question:
The following data were generated from an $N(\mu, 1)$ distribution by a student. Unfortunately, the student forgot which value of $\mu$ was used, so we are uncertain about the correct probability distribution to use to describe the variation in the data.

\begin{tabular}{rrrrrrrr|}
\hline 0.2 & -0.7 & 0.0 & -1.9 & 0.7 & -0.3 & 0.3 & 0.4 \\
0.3 & -0.8 & 1.5 & 0.1 & 0.3 & -0.7 & -1.8 & 0.2 \\
\hline
\end{tabular}

Can you suggest a plausible value for $\mu$ ? Explain your reasoning.

Answer: $\bar{x}=-0.1375$

The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.


Question: Suppose you are measuring the volume of a cubic box in centimeters by taking repeated independent measurements of one of the sides. Suppose it is reasonable to assume that a single measurement follows an $N\left(\mu, \sigma_{0}^{2}\right)$ distribution, where $\mu$ is unknown and $\sigma_{0}^{2}$ is known. Based on a sample of measurements, you obtain the MLE of $\mu$ as 3.2 $\mathrm{cm}$. What is your estimate of the volume of the box? How do you justify this in terms of the likelihood function?

Answer: $\hat{\mu}^{3}=32.768 \mathrm{~cm}^{3}$ is the MLE

The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.


Question: Suppose that a simple linear model is fit to data. An analysis of the residuals indicates that there is no reason to doubt that the model is correct; the ANOVA test indicates that there is substantial evidence against the null hypothesis of no relationship between the response and predictor. The value of $R^{2}$ is found to be 0.05 . What is the interpretation of this number and what are the practical consequences?

Answer: $R^{2}=0.05$ indicates that the linear model explains only $5 \%$ of the variation in the response, so the model will not have much predictive power.

The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.


Question: Suppose there are three coins - one is known to be fair, one has probability $1 / 3$ of yielding a head on a single toss, and one has probability $2 / 3$ for head on a single toss. A coin is selected (not randomly) and then tossed five times. The goal is to make an inference about which of the coins is being tossed, based on the sample. Fully describe a statistical model for a single response and for the sample.

Answer: The statistical model for a single response consists of three probability functions \{Bernoulli(1/2), Bernoulli(1/3), Bernoulli(2/3)\}.

The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.


Question: Let $X \sim \operatorname{Binomial}(80,1 / 4)$, and let $Y \sim \operatorname{Poisson}(3 / 2)$. Assume $X$ and $Y$ are independent. Compute $E(X Y)$.

Answer: $E(X Y)=30$

The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.


Question: Marks on an exam in a statistics course are assumed to be normally distributed with unknown mean but with variance equal to 5 . A sample of four students is selected, and their marks are $52,63,64,84$. Assess the hypothesis $H_{0}: \mu=60$ by computing the relevant $\mathrm{P}$-value and compute a 0.95 -confidence interval for the unknown $\mu$.

Answer: P-value $=0.000$ and 0.95 -confidence interval is $(63.56,67.94)$.

The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.


Question: Suppose two measurements, $Y_{1}$ and $Y_{2}$, corresponding to different treatments, are taken on the same individual who has been randomly sampled from a population $\Pi$. Suppose that $Y_{1}$ and $Y_{2}$ have the same variance and are negatively correlated. Our goal is to compare the treatment means. Explain why it would have been better to have randomly sampled two individuals from $\Pi$ and applied the treatments to these individuals separately. (Hint: Consider $\operatorname{Var}\left(Y_{1}-Y_{2}\right)$ in these two sampling situations.)

Answer: When $Y_{1}$ and $Y_{2}$ are measured on the same individual, we have that $\operatorname{Var}\left(Y_{1}-\right.$ $\left.Y_{2}\right)=2\left(\operatorname{Var}\left(Y_{1}\right)-\operatorname{Cov}\left(Y_{1}, Y_{2}\right)\right)>2 \operatorname{Var}\left(Y_{1}\right)$ since $\operatorname{Cov}\left(Y_{1}, Y_{2}\right)<0$. If we had measured $Y_{1}$ and $Y_{2}$ on independently randomly selected individuals, then we would have that $\operatorname{Var}\left(Y_{1}-Y_{2}\right)=2 \operatorname{Var}\left(Y_{1}\right)$.

The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.


Question: Suppose $S=\{1,2,3\}$, with $P(\{1\})=1 / 2$ and $P(\{1,2\})=2 / 3$. What must $P(\{2\})$ be?

Answer: $P(\{2\})=1 / 6$

The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.


Question: Suppose a student wants to put a prior on the mean grade out of 100 that their class will obtain on the next statistics exam. The student feels that a normal prior centered at 66 is appropriate and that the interval $(40,92)$ should contain $99 \%$ of the marks. Fully identify the prior.

Answer: The prior distribution is $\theta \sim N\left(66, \sigma^{2}\right)$ with $\sigma^{2}=101.86$.

The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.


Question: Suppose the cdf of $W$ is given by $F_{W}(w)=0$ for $w<10, F_{W}(w)=w-10$ for $10 \leq w \leq 11$, and by $F_{W}(w)=1$ for $w>11$. Compute $E(W)$. (Hint: Remember that $F_{W}(w)=P(W \leq w)=1-P(W>w)$.)

Answer: $E(W)=21 / 2$

The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.


Question: Suppose that $\left(x_{1}, \ldots, x_{n}\right)$ is a sample from an $N\left(\mu, \sigma_{0}^{2}\right)$ distribution, where $\mu \in$ $R^{1}$ is unknown and $\sigma_{0}^{2}$ is known. Determine a UMVU estimator of the second moment $\mu^{2}+\sigma_{0}^{2}$.

Answer: $\bar{x}^{2}+(1-1 / n) \sigma_{0}^{2}$

The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.


Question: Let $W \sim \operatorname{Binomial}(100,1 / 2)$, as in the number of heads when flipping 100 fair coins. Use Chebychev's inequality to get an upper bound on $P(|W-50| \geq 10)$.

Answer: $1 / 4$

The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.


Question: A P-value was computed to assess the hypothesis $H_{0}: \psi(\theta)=0$ and the value 0.22 was obtained. The investigator says this is strong evidence that the hypothesis is correct. How do you respond?

Answer: The P-value 0.22 does not imply the null hypothesis is correct. It may be that we have just not taken a large enough sample size to detect a difference.

The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.


Question: Let $\{N(t)\}_{t \geq 0}$ be a Poisson process with intensity $a=1 / 3$. Compute $P\left(N_{2}=\right.$ 6) and $P\left(N_{3}=5\right)$.

Answer: $P\left(N_{2}=6\right)=e^{-2 / 3}(2 / 3)^{6} / 6$ !, $P\left(N_{3}=5\right)=e^{-3 / 3}(3 / 3)^{5} / 5$ !

The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.


Question: Suppose that $X \sim \operatorname{Geometric}(1 / 3)$. What value would you record as a prediction of a future value of $X$ ?

Answer: The mode is $x=0$.

The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.


Question: If two functions of $\theta$ are equivalent versions of the likelihood when one is a positive multiple of the other, then when are two log-likelihood functions equivalent?

Answer: Equivalent log-likelihood functions differ by an additive constant.

The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.


Question: Suppose that a statistical model is given by the family of $\operatorname{Bernoulli}(\theta)$ distributions where $\theta \in \Omega=[0,1]$. If our interest is in making inferences about the probability that in two independent observations from this model we obtain a 0 and a 1 , then determine $\psi(\theta)$.

Answer: $\psi(\theta)=2 \theta(1-\theta)$

The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.


Question: Suppose a certain experiment has probability $\theta$ of success, where $0<\theta<1$ but $\theta$ is unknown. Suppose we repeat the experiment 1000 times, of which 400 are successes and 600 are failures. Compute an interval of values that are virtually certain to contain $\theta$.

Answer: $(0.354,0.447)$

The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.


Question: Suppose a university is composed of $55 \%$ female students and $45 \%$ male students. A student is selected to complete a questionnaire. There are 25 questions on the questionnaire administered to a male student and 30 questions on the questionnaire administered to a female student. If $X$ denotes the number of questions answered by a randomly selected student, then compute $P(X=x)$ for every real number $x$.

Answer: $P(X=25)=0.45, P(X=30)=0.55$, and $P(X=x)=0$ otherwise

The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.


Question: Let $A$ and $B$ be events, and let $X=I_{A} \cdot I_{B}$. Is $X$ an indicator function? If yes, then of what event?

Answer: Yes, for $A \cap B$.

The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.


Question: Suppose a possibly biased die is rolled 30 times and that the face containing two pips comes up 10 times. Do we have evidence to conclude that the die is biased?

Answer: P-value $=0.014$

The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.


Question: Suppose that $\left(x_{1}, \ldots, x_{n}\right)$ is a sample from an $N\left(\mu, \sigma_{0}^{2}\right)$ distribution, where $\mu \in$ $R^{1}$ is unknown and $\sigma_{0}^{2}$ is known. Is $2 \bar{x}+3$ a UMVU estimator of anything? If so, what is it UMVU for? Justify your answer.

Answer: UMVU for $5+2 \mu$

The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.


Question: Suppose your team has a $40 \%$ chance of winning or tying today's game and has a $30 \%$ chance of winning today's game. What is the probability that today's game will be a tie?

Answer: $10 \%$

The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.


Question: Suppose a species of beetle has length 35 millimeters on average. Find an upper bound on the probability that a randomly chosen beetle of this species will be over 80 millimeters long.

Answer: $7 / 16$

The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.


Question: Suppose that a statistical model is given by the family of $N\left(\mu, \sigma_{0}^{2}\right)$ distributions where $\theta=\mu \in R^{1}$ is unknown, while $\sigma_{0}^{2}$ is known. If our interest is in making inferences about the distribution function evaluated at 3, then determine $\psi(\mu)$.

Answer: $\psi(\mu)=\Phi\left((3-\mu) / \sigma_{0}\right)$

The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.


Question: Suppose that $X$ is a random variable and $Y=X^{2}$. Determine whether or not $X$ and $Y$ are related. What happens when $X$ has a degenerate distribution?

Answer: The conditional distributions $P(Y=y \mid X=x)$ will change with $x$ whenever $X$ is not degenerate.

The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.


Question: Suppose that $X \sim N(10,2)$. What value would you record as a prediction of a future value of $X$ ? How would you justify your choice?

Answer: $x=10$

The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.


Question: Suppose measurements (in centimeters) are taken using an instrument. There is error in the measuring process and a measurement is assumed to be distributed $N\left(\mu, \sigma_{0}^{2}\right)$, where $\mu$ is the exact measurement and $\sigma_{0}^{2}=0.5$. If the $(n=10)$ measurements 4.7, 5.5, 4.4, 3.3, 4.6, 5.3, 5.2, 4.8, 5.7, 5.3 were obtained, assess the hypothesis $H_{0}: \mu=5$ by computing the relevant P-value. Also compute a 0.95 -confidence interval for the unknown $\mu$.

Answer: $\mathrm{P}$-value $=0.592$ and 0.95 -confidence interval is $(4.442,5.318)$.

The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.


Question: Suppose a variable $X$ takes the values 1 and 2 on a population and the conditional distributions of $Y$ given $X$ are $N(0,5)$ when $X=1$, and $N(0,7)$ when $X=2$. Determine whether $X$ and $Y$ are related and if so, describe their relationship.

Answer: $X$ and $Y$ are related. We see that only the variance of the conditional distribution changes as we change $X$.

The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.


Question: Suppose that $\left(x_{1}, \ldots, x_{n}\right)$ is a sample from the Exponential $(\theta)$, where $\theta>0$ is unknown. What is the least-squares estimate of the mean of this distribution?

Answer: $\bar{x}$

The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.


Question: Consider flipping two independent fair coins. Let $X$ be the number of heads that appear. Compute $P(X=x)$ for all real numbers $x$.

Answer: $P(X=0)=P(X=2)=1 / 4, P(X=1)=1 / 2, P(X=x)=0$ for $x \neq 0,1,2$

The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.


Question: Let $X \sim \operatorname{Binomial}(12, \theta)$. For what value of $\theta$ is $P(X=11)$ maximized?

Answer: $\theta=11 / 12$

The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.


Question: Is the function defined by $f(x)=x / 3$ for $-1<x<2$ and 0 otherwise, a density? Why or why not?

Answer: No

The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.


Question:
Suppose we have a sample

\begin{tabular}{|llllllllll|}
\hline 6.56 & 6.39 & 3.30 & 3.03 & 5.31 & 5.62 & 5.10 & 2.45 & 8.24 & 3.71 \\
4.14 & 2.80 & 7.43 & 6.82 & 4.75 & 4.09 & 7.95 & 5.84 & 8.44 & 9.36 \\
\hline
\end{tabular}

from an $N\left(\mu, \sigma^{2}\right)$ distribution and we determine that a prior specified by $\mu \mid \sigma^{2} \sim$ $N\left(3,4 \sigma^{2}\right), \sigma^{-2} \sim \operatorname{Gamma}(1,1)$ is appropriate. Determine the posterior distribution of $\left(\mu, 1 / \sigma^{2}\right)$.

Answer: $\mu\left|\sigma^{2}, x_{1}, \ldots, x_{n} \sim N\left(5.5353, \frac{4}{81} \sigma^{2}\right), 1 / \sigma^{2}\right| x_{1}, \ldots, x_{n} \sim \operatorname{Gamma}(11,41.737)$

The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.


Question: Suppose that an employee arrives late $10 \%$ of the time, leaves early $20 \%$ of the time, and both arrives late and leaves early $5 \%$ of the time. What is the probability that on a given day that employee will either arrive late or leave early (or both)?

Answer: $P$ (late or early or both) $=25 \%$

The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.


Question: Write out in full how you would generate a value from a Dirichlet $(1,1,1,1)$ distribution.

Answer: We should first generate a value for $X_{1} \sim \operatorname{Dirichlet}(1,3)$. Then generate $U_{2}$ from the $\operatorname{Beta}(1,2)$ distribution and set $X_{2}=\left(1-X_{1}\right) U_{2}$. Next generate $U_{3}$ from the $\operatorname{Beta}(1,1)$ distribution and set $X_{3}=\left(1-X_{1}-X_{2}\right) U_{3}$. Finally, set $X_{4}=1-X_{1}-$ $X_{2}-X_{3}$.

The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.


Question: If $\left(x_{1}, \ldots, x_{n}\right)$ is a sample from a $\operatorname{Bernoulli}(\theta)$ distribution, where $\theta \in[0,1]$ is unknown, then determine the MLE of $\theta^{2}$.

Answer: $\psi(\theta)=\theta^{2}$ is $1-1$, and so $\psi\left(\hat{\theta}\left(x_{1}, \ldots, x_{n}\right)\right)=\bar{x}^{2}$ is the MLE.

The book is copyright (c) by Michael J. Evans and Jeffrey S. Rosenthal. It may be copied and distributed without restriction, provided it is not altered, appropriate attribution is given and no money is charged.


Question:
Determine if the system is consistent. If so, is the solution unique?
$$
\begin{gathered}
x+2 y+z-w=2 \\
x-y+z+w=1 \\
2 x+y-z=1 \\
4 x+2 y+z=5
\end{gathered}
$$

Answer: There is no solution.

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Question: Suppose that $A, B, C, D$ are $n \times n$-matrices, and that all relevant matrices are invertible. Further, suppose that $(A+B)^{-1}=C B^{-1}$. Solve this equation for $A$ (in terms of $B$ and $C$ ), $B$ (in terms of $A$ and $C$ ), and $C$ (in terms of $A$ and $B$ ).

Answer:
To solve for $A$, we invert both sides of the equation $(A+B)^{-1}=C B^{-1}$ and use matrix algebra to get $A+B=\left(C B^{-1}\right)^{-1}=\left(B^{-1}\right)^{-1} C^{-1}=B C^{-1}$. Therefore, $A=B C^{-1}-B$.

To solve for $B$, we note that $A=B C^{-1}-B=B\left(C^{-1}-I\right)$. Multiplying both sides of the equation on the right by the inverse of $C^{-1}-I$, we get $B=A\left(C^{-1}-I\right)^{-1}$.

To solve for $C$, we take the original equation $(A+B)^{-1}=C B^{-1}$ and right-multiply both sides of the equation $B$. This yields $C=(A+B)^{-1} B$.

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Question:
Are the following vectors linearly independent? If not, write one of them as a linear combination of the others.

$$
\mathbf{u}=\left[\begin{array}{l}
1 \\
3 \\
1
\end{array}\right], \quad \mathbf{v}=\left[\begin{array}{l}
1 \\
4 \\
2
\end{array}\right], \quad \mathbf{w}=\left[\begin{array}{r}
1 \\
1 \\
-1
\end{array}\right]
$$

Answer: The vectors are linearly dependent. We have $\mathbf{w}=3 \mathbf{u}-2 \mathbf{v}$.

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Question: Simplify $\|\mathbf{u} \times \mathbf{v}\|^{2}+(\mathbf{u} \cdot \mathbf{v})^{2}-\|\mathbf{u}\|^{2}\|\mathbf{v}\|^{2}$.

Answer:
We have

$$
\begin{aligned}
\|\mathbf{u} \times \mathbf{v}\|^{2} & =\|\mathbf{u}\|^{2}\|\mathbf{v}\|^{2} \sin ^{2} \theta \\
& =\|\mathbf{u}\|^{2}\|\mathbf{v}\|^{2}\left(1-\cos ^{2} \theta\right) \\
& =\|\mathbf{u}\|^{2}\|\mathbf{v}\|^{2}-\|\mathbf{u}\|^{2}\|\mathbf{v}\|^{2} \cos ^{2} \theta \\
& =\|\mathbf{u}\|^{2}\|\mathbf{v}\|^{2}-(\mathbf{u} \cdot \mathbf{v})^{2}
\end{aligned}
$$

which implies the expression equals 0 .

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Question: Find the area of the parallelogram determined by the vectors $\left[\begin{array}{l}1 \\ 0 \\ 3\end{array}\right],\left[\begin{array}{r}4 \\ -2 \\ 1\end{array}\right]$.

Answer: $\left[\begin{array}{l}1 \\ 0 \\ 3\end{array}\right] \times\left[\begin{array}{r}4 \\ -2 \\ 1\end{array}\right]=\left[\begin{array}{r}6 \\ 11 \\ -2\end{array}\right]$. The area is of the parallelogram is $\sqrt{36+121+4}=\sqrt{161}$.

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Question: Suppose a system of equations has fewer equations than variables and you have found a solution to this system of equations. Is it possible that your solution is the only one? Explain.

Answer: No. Consider $x+y+z=2$ and $x+y+z=1$.

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Question:
Which of the following matrices are symmetric, antisymmetric, both, or neither?
$$
A=\left[\begin{array}{rr}
0 & 1 \\
-1 & 0
\end{array}\right], \quad B=\left[\begin{array}{ll}
2 & 1 \\
1 & 3
\end{array}\right], \quad C=\left[\begin{array}{rr}
1 & 2 \\
-2 & 0
\end{array}\right], \quad D=\left[\begin{array}{ll}
0 & 0 \\
0 & 0
\end{array}\right] .

$$

Answer: $A$ is antisymmetric, $B$ is symmetric, $C$ is neither, and $D$ is both.

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Question:
Find the eigenvalues and eigenvectors of the matrix
$$
\left[\begin{array}{rrr}
-13 & -28 & 28 \\
4 & 9 & -8 \\
-4 & -8 & 9
\end{array}\right]
$$
One eigenvalue is 3. Diagonalize if possible.

Answer:
The eigenvectors and eigenvalues are:

$$
\left\{\left[\begin{array}{l}
2 \\
0 \\
1
\end{array}\right],\left[\begin{array}{c}
-2 \\
1 \\
0
\end{array}\right]\right\} \text { for eigenvalue } 1, \quad\left\{\left[\begin{array}{c}
7 \\
-2 \\
2
\end{array}\right]\right\} \text { for eigenvalue } 3 .
$$

The matrix $P$ needed to diagonalize the above matrix is

$$
\left[\begin{array}{rrr}
2 & -2 & 7 \\
0 & 1 & -2 \\
1 & 0 & 2
\end{array}\right]
$$

and the diagonal matrix $D$ is

$$
\left[\begin{array}{lll}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 3
\end{array}\right]
$$

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Question:
Here are some vectors.

$$
\mathbf{u}_{1}=\left[\begin{array}{r}
1 \\
2 \\
-2
\end{array}\right], \quad \mathbf{u}_{2}=\left[\begin{array}{r}
2 \\
2 \\
-4
\end{array}\right], \quad \mathbf{u}_{3}=\left[\begin{array}{r}
2 \\
7 \\
-4
\end{array}\right], \quad \mathbf{u}_{4}=\left[\begin{array}{r}
5 \\
7 \\
-10
\end{array}\right], \quad \mathbf{u}_{5}=\left[\begin{array}{r}
12 \\
17 \\
-24
\end{array}\right] .
$$

Describe the span of these vectors as the span of as few vectors as possible.

Answer:
$$
\left[\begin{array}{rrrrr}
1 & 2 & 2 & 5 & 12 \\
1 & 2 & 7 & 7 & 17 \\
-2 & -4 & -4 & -10 & -24
\end{array}\right] \simeq\left[\begin{array}{llllr}
1 & 1 & 2 & 5 & 12 \\
0 & 0 & 5 & 2 & 5 \\
0 & 0 & 0 & 0 & 0
\end{array}\right]
$$
Linearly independent subset: $\left\{\mathbf{u}_{1}, \mathbf{u}_{3}\right\}$. Since the rank is 2 , this is the smallest possible.

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Question:
What are the dimensions of the following matrices?
$$
A=\left[\begin{array}{rrr}
1 & -2 & 0 \\
4 & 3 & 2
\end{array}\right], \quad B=\left[\begin{array}{lll}
3 & 4 & 1 \\
1 & 3 & 1 \\
6 & 2 & 2
\end{array}\right], \quad C=\left[\begin{array}{ll}
1 & 0 \\
4 & 0 \\
2 & 0 \\
0 & 0
\end{array}\right]
$$

Answer: $2 \times 3,3 \times 3,4 \times 2$.

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Question:
Consider the following vector equation for a line in $\mathbb{R}^{3}$ :
$$
\left[\begin{array}{l}
x \\
y \\
z
\end{array}\right]=\left[\begin{array}{l}
1 \\
2 \\
0
\end{array}\right]+t\left[\begin{array}{l}
1 \\
0 \\
1
\end{array}\right]
$$
Find a new vector equation for the same line by doing the change of parameter $t=2-s$.

Answer:
We have

$$
\left[\begin{array}{l}
x \\
y \\
z
\end{array}\right]=\left[\begin{array}{l}
1 \\
2 \\
0
\end{array}\right]+(2-s)\left[\begin{array}{l}
1 \\
0 \\
1
\end{array}\right]=\left[\begin{array}{l}
3 \\
2 \\
2
\end{array}\right]+s\left[\begin{array}{r}
-1 \\
0 \\
-1
\end{array}\right] .
$$

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Question: Use Gauss-Jordan elimination to solve the system of equations $-8 x+2 y+5 z=18,-8 x+$ $3 y+5 z=13$, and $-4 x+y+5 z=19$.

Answer: Solution is: $[x=-1, y=-5, z=4]$

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Question:
Which of the following vectors, if any, are orthogonal?
$\mathbf{u}_{1}=\left[\begin{array}{c}1 \\ i \\ 0\end{array}\right]$,
$\mathbf{u}_{2}=\left[\begin{array}{c}1 \\ -i \\ 1\end{array}\right]$,
$\mathbf{u}_{3}=\left[\begin{array}{c}0 \\ 1 \\ -i\end{array}\right]$,
$\mathbf{u}_{4}=\left[\begin{array}{c}0 \\ 0 \\ 1\end{array}\right]$.

Answer: $\mathbf{u}_{1} \perp \mathbf{u}_{2}, \mathbf{u}_{1} \perp \mathbf{u}_{4}$, and $\mathbf{u}_{2} \perp \mathbf{u}_{3}$.

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Question:
Compute $A^{4}$, where
$$
A=\left[\begin{array}{ll}
1 & -3 \\
2 & -5
\end{array}\right]
$$
Hint: you can save some work by calculating $A^{2}$ times $A^{2}$.

Answer: $\left[\begin{array}{rr}-71 & 168 \\ -112 & 265\end{array}\right]$

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Question: Unitarily diagonalize the hermitian matrix $A=\left[\begin{array}{cc}2 & 1+i \\ 1-i & 3\end{array}\right]$.

Answer:
Eigenvalues: $\lambda_{1}=1$ and $\lambda_{2}=4$. Normalized eigenvectors: $\mathbf{v}_{1}=\frac{1}{\sqrt{3}}\left[\begin{array}{c}1+i \\ -1\end{array}\right], \mathbf{v}_{2}=\frac{1}{\sqrt{3}}\left[\begin{array}{c}1 \\ 1-i\end{array}\right]$.

$$
D=\left[\begin{array}{ll}
1 & 0 \\
0 & 4
\end{array}\right], \quad P=\frac{1}{\sqrt{3}}\left[\begin{array}{cc}
1+i & 1 \\
-1 & 1-i
\end{array}\right] \text {. }
$$`

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Question: Consider the points $\left(x_{1}, y_{1}\right)=(-1,4),\left(x_{2}, y_{2}\right)=(0,-2),\left(x_{3}, y_{3}\right)=(1,4),\left(x_{4}, y_{4}\right)=$ $(2,2)$. Find the least squares parabola for these points.

Answer: $y=1-x+x^{2}$.

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Question: Find the area of the parallelogram with vertices $(-2,3,1),(2,1,1),(1,2,-1)$, and $(5,0,-1)$.

Answer: Let $P=(-2,3,1), Q=(2,1,1), R=(1,2,-1)$, and $S=(5,0,-1)$. We have $\overrightarrow{P Q} \times \overrightarrow{P R}=\left[\begin{array}{r}4 \\ -2 \\ 0\end{array}\right] \times$ $\left[\begin{array}{r}3 \\ -1 \\ -2\end{array}\right]=\left[\begin{array}{l}5 \\ 8 \\ 2\end{array}\right]$. The area of the parallelogram is $\|\overrightarrow{P Q} \times \overrightarrow{P R}\|=\sqrt{5^{2}+8^{2}+2^{2}}=\sqrt{93}$

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Question:
Assume $\mathbf{u}, \mathbf{v}, \mathbf{w}$ are linearly independent elements of some vector space $V$. Consider the set of vectors
$$
R=\left\{2 \mathbf{u}-\mathbf{w}, \mathbf{w}+\mathbf{v}, 3 \mathbf{v}+\frac{1}{2} \mathbf{u}\right\}
$$
Determine whether $R$ is linearly independent.

Answer: Therefore, the set $R$ is linearly independent.

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Question:
Which of the following vectors are redundant? If there are redundant vectors, write each of them as a linear combination of previous vectors.

$$
\mathbf{u}_{1}=\left[\begin{array}{l}
1 \\
0 \\
1
\end{array}\right], \quad \mathbf{u}_{2}=\left[\begin{array}{l}
2 \\
0 \\
2
\end{array}\right], \quad \mathbf{u}_{3}=\left[\begin{array}{l}
1 \\
2 \\
1
\end{array}\right], \quad \mathbf{u}_{4}=\left[\begin{array}{l}
1 \\
6 \\
1
\end{array}\right]
$$

Answer: $\mathbf{u}_{2}$ and $\mathbf{u}_{4}$ are redundant. We have $\mathbf{u}_{2}=2 \mathbf{u}_{1}$ and $\mathbf{u}_{4}=3 \mathbf{u}_{3}-2 \mathbf{u}_{1}$.

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Question:
Find a linearly independent set of vectors that has the same span as the given vectors.

$$
\mathbf{u}_{1}=\left[\begin{array}{l}
2 \\
0 \\
3
\end{array}\right], \quad \mathbf{u}_{2}=\left[\begin{array}{l}
1 \\
3 \\
5
\end{array}\right], \quad \mathbf{u}_{3}=\left[\begin{array}{l}
3 \\
3 \\
8
\end{array}\right], \quad \mathbf{u}_{4}=\left[\begin{array}{r}
3 \\
-3 \\
1
\end{array}\right]
$$

Answer: $\left[\begin{array}{rrrr}2 & 1 & 3 & 3 \\ 0 & 3 & 3 & -3 \\ 3 & 5 & 8 & 1\end{array}\right] \simeq\left[\begin{array}{rrrr}1 & 4 & 5 & -2 \\ 0 & 1 & 1 & -1 \\ 0 & 0 & 0 & 0\end{array}\right]$. Linearly independent subset: $\left\{\mathbf{u}_{1}, \mathbf{u}_{2}\right\}$.

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Question:
Which of the following are unit vectors?
$$
\mathbf{u}=\frac{1}{2}\left[\begin{array}{l}
1 \\
1 \\
1
\end{array}\right], \quad \mathbf{v}=\frac{1}{3}\left[\begin{array}{l}
1 \\
1 \\
1
\end{array}\right], \quad \mathbf{w}=\frac{1}{2}\left[\begin{array}{l}
1 \\
1 \\
1 \\
1
\end{array}\right]
$$

Answer: Only $\mathbf{w}$ is a unit vector.

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Question: Find $\operatorname{proj}_{\mathbf{v}}(\mathbf{w})$ where $\mathbf{w}=\left[\begin{array}{r}1 \\ 2 \\ -2\end{array}\right]$ and $\mathbf{v}=\left[\begin{array}{l}1 \\ 0 \\ 3\end{array}\right]$.

Answer: $\frac{\mathbf{v} \cdot \mathbf{w}}{\mathbf{v} \cdot \mathbf{v}} \mathbf{v}=\frac{-5}{10}\left[\begin{array}{l}1 \\ 0 \\ 3\end{array}\right]=\left[\begin{array}{c}-\frac{1}{2} \\ 0 \\ -\frac{3}{2}\end{array}\right]$

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Question:
Decompose the vector $\mathbf{v}$ into $\mathbf{v}=\mathbf{a}+\mathbf{b}$ where $\mathbf{a}$ is parallel to $\mathbf{u}$ and $\mathbf{b}$ is orthogonal to $\mathbf{u}$.
$$
\mathbf{v}=\left[\begin{array}{r}
3 \\
2 \\
-5
\end{array}\right], \quad \mathbf{u}=\left[\begin{array}{r}
1 \\
-1 \\
2
\end{array}\right]
$$

Answer:
$$
\mathbf{a}=\left[\begin{array}{c}
-3 / 2 \\
3 / 2 \\
-3
\end{array}\right], \quad \mathbf{b}=\left[\begin{array}{c}
9 / 2 \\
1 / 2 \\
-2
\end{array}\right]
$$

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Question: Can a column vector ever be equal to a row vector?

Answer: Yes, a $1 \times 1$-matrix is both a row vector and a column vector. However, a column vector of dimension 2 or greater can never be equal to a row vector, because one is an $n \times 1$-matrix and the other is a $1 \times n$ matrix.

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Question:
In $\mathbb{R}^{3}$ with the usual dot product, find an orthogonal basis for
$$
\operatorname{span}\left\{\left[\begin{array}{l}
1 \\
2 \\
3
\end{array}\right],\left[\begin{array}{l}
2 \\
6 \\
0
\end{array}\right]\right\}
$$

Answer: $\mathbf{u}_{1}=\left[\begin{array}{l}1 \\ 2 \\ 3\end{array}\right], \mathbf{u}_{2}=\left[\begin{array}{c}1 \\ 4 \\ -3\end{array}\right]$

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Question: Let $z=2+7 i$ and let $w=3-8 i$. Compute $z+w, z-2 w$, $z w$, and $\frac{w}{z}$.

Answer: $z+w=5-i, z-2 w=-4+23 i, z w=62+5 i$, and $\frac{w}{z}=-\frac{50}{53}-\frac{37}{53} i$.

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Question:
Find $h$ such that
$$
\left[\begin{array}{ll|l}
1 & h & 3 \\
2 & 4 & 6
\end{array}\right]
$$
is the augmented matrix of a consistent system.

Answer: Any $h$ will work.

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Question: Use elementary operations to find the point $(x, y)$ that lies on both lines $x+3 y=1$ and $4 x-y=3$.

Answer: Solution is: $(x, y)=\left(\frac{10}{13}, \frac{1}{13}\right)$

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Question:
Describe the span of the following vectors in $\mathbb{R}^{4}$:
$$
\mathbf{u}_{1}=\left[\begin{array}{r}
1 \\
1 \\
-1 \\
-1
\end{array}\right], \quad \mathbf{u}_{2}=\left[\begin{array}{l}
5 \\
1 \\
1 \\
0
\end{array}\right], \quad \mathbf{u}_{3}=\left[\begin{array}{r}
0 \\
-2 \\
2 \\
1
\end{array}\right], \quad \mathbf{u}_{4}=\left[\begin{array}{l}
3 \\
1 \\
1 \\
1
\end{array}\right]
$$

Answer: It is the hyperplane given by the equation $x-2 y-3 z+2 w=0$.

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Question: Find $\left[\begin{array}{l}1 \\ 2 \\ 3\end{array}\right]+\left[\begin{array}{l}1 \\ 5 \\ 1\end{array}\right]+\left[\begin{array}{c}-1 \\ 2 \\ -4\end{array}\right]$.

Answer: $\left[\begin{array}{l}1 \\ 9 \\ 0\end{array}\right]$.

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Question: My system of equations has a solution $(x, y, z)=(1,2,4)$. The associated homogeneous system has basic solutions $(x, y, z)=(1,0,1)$ and $(x, y, z)=(0,1,-1)$. What is the general solution of my system of equations?

Answer: The general solution is $(x, y, z)=(1,2,4)+s(1,0,1)+t(0,1,-1)$, or equivalently, $(x, y, z)=(1+$ $s, 2+t, 4+s-t)$, where $s$ and $t$ are parameters.

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Question:
Find the eigenvalues and eigenvectors of the matrix
$$
\left[\begin{array}{rrr}
5 & -18 & -32 \\
0 & 5 & 4 \\
2 & -5 & -11
\end{array}\right]
$$
One eigenvalue is 1. Diagonalize if possible.

Answer:
The eigenvalues are -1 and 1. The eigenvectors corresponding to the eigenvalues are:

$$
\left\{\left[\begin{array}{c}
10 \\
-2 \\
3
\end{array}\right]\right\} \text { for eigenvalue }-1, \quad\left\{\left[\begin{array}{c}
7 \\
-2 \\
2
\end{array}\right]\right\} \text { for eigenvalue } 1 .
$$

Since there are only 2 linearly independent eigenvectors, this matrix is not diagonalizable.

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Question:
Find the length of each of the following vectors.
$$
\mathbf{u}=\left[\begin{array}{c}
-3 \\
2
\end{array}\right], \quad \mathbf{v}=\left[\begin{array}{c}
1 \\
-2 \\
5
\end{array}\right], \quad \mathbf{w}=\left[\begin{array}{c}
1 \\
4 \\
-2 \\
1
\end{array}\right]
$$

Answer: $\|\mathbf{u}\|=\sqrt{13},\|\mathbf{v}\|=\sqrt{30},\|\mathbf{w}\|=\sqrt{22}$.

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Question: Let $A=\left[\begin{array}{ll}1 & 2 \\ 3 & 4\end{array}\right]$ and $B=\left[\begin{array}{ll}1 & 2 \\ 3 & k\end{array}\right]$. Is it possible to find $k$ such that $A B=B A$? If so, what should $k$ equal?

Answer: Solution is: $k=4$.

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Question:
Decrypt the message "ERM DXYBJUWW. JWQLD, HL" using the Hill cipher with block size 3 and encryption matrix
$$
A=\left[\begin{array}{lll}
2 & 1 & 1 \\
1 & 3 & 1 \\
1 & 1 & 4
\end{array}\right]
$$

Answer:
The decryption matrix is

$$
A^{-1}=\left[\begin{array}{ccc}
16 & 22 & 5 \\
22 & 26 & 17 \\
5 & 17 & 2
\end{array}\right]
$$

Plaintext: "Spies are at the gate".

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Question: Find $-3\left[\begin{array}{r}5 \\ -1 \\ 2 \\ -3\end{array}\right]+5\left[\begin{array}{r}-8 \\ 2 \\ -3 \\ 6\end{array}\right]$.

Answer: $\left[\begin{array}{r}-55 \\ 13 \\ -21 \\ 39\end{array}\right]$.

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Question:
Let
$$
A=\left[\begin{array}{ccc}
3 & -1 & 0 \\
-1 & 5 & 2 \\
0 & 2 & 3
\end{array}\right]
$$
and consider the vector space $\mathbb{R}^{3}$ with the inner product given by $\langle\mathbf{v}, \mathbf{w}\rangle=\mathbf{v}^{T}$ Aw. Let
$$
\mathbf{v}_{1}=\left[\begin{array}{l}
1 \\
0 \\
2
\end{array}\right], \quad \mathbf{v}_{2}=\left[\begin{array}{c}
-1 \\
1 \\
-5
\end{array}\right], \quad \text { and } \quad \mathbf{v}_{3}=\left[\begin{array}{l}
2 \\
2 \\
3
\end{array}\right]
$$
Apply the Gram-Schmidt procedure to $\mathbf{v}_{1}, \mathbf{v}_{2}, \mathbf{v}_{3}$ to find an orthogonal basis $\left\{\mathbf{u}_{1}, \mathbf{u}_{2}, \mathbf{u}_{3}\right\}$ for $\mathbb{R}^{3}$ with respect to the above inner product.

Answer: $\mathbf{u}_{1}=\left[\begin{array}{l}1 \\ 0 \\ 2\end{array}\right], \mathbf{u}_{2}=\left[\begin{array}{c}1 \\ 1 \\ -1\end{array}\right], \mathbf{u}_{3}=\left[\begin{array}{c}-1 \\ 1 \\ 0\end{array}\right]$

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Question:
Eve intercepts the following encrypted message sent by Bob:
“TGVXKHGSW, JU, JHY JSCDSBQIRPEV”
Eve knows that Alice uses a Hill cipher with block length 2, but she does not know the secret encryption matrix. Eve also knows that Bob begins all of his letters with "Hello". Decrypt the message.

Answer:
The first two plaintext blocks are $(8,5),(12,12)$ and the first two ciphertext blocks are $(20,7),(22,24)$. Eve solves the equation

$$
A^{-1}\left[\begin{array}{cc}
20 & 22 \\
7 & 24
\end{array}\right]=\left[\begin{array}{ll}
8 & 12 \\
5 & 12
\end{array}\right]
$$

to find the secret decryption matrix

$$
A^{-1}=\left[\begin{array}{ll}
3 & 5 \\
1 & 2
\end{array}\right] .
$$

The plaintext is "Hello, password is kiwifruit".

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Question: Suppose $B=\left\{\mathbf{u}_{1}, \mathbf{u}_{2}, \mathbf{u}_{3}\right\}$ is an orthogonal basis for an inner product space $V$, such that $\left\|\mathbf{u}_{1}\right\|=2,\left\|\mathbf{u}_{2}\right\|=\sqrt{3}$, and $\left\|\mathbf{u}_{3}\right\|=\sqrt{5}$. Moreover, suppose that $\mathbf{v} \in V$ is a vector such that $\left\langle\mathbf{v}, \mathbf{u}_{1}\right\rangle=1$, $\left\langle\mathbf{v}, \mathbf{u}_{2}\right\rangle=2$, and $\left\langle\mathbf{v}, \mathbf{u}_{3}\right\rangle=-4$. Find the coordinates of $\mathbf{v}$ with respect to $B$.

Answer: $\mathbf{v}=\frac{1}{4} \mathbf{u}_{1}+\frac{2}{3} \mathbf{u}_{2}-\frac{4}{5} \mathbf{u}_{3}$

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Question: Find the volume of the parallelepiped determined by the vectors $\left[\begin{array}{r}1 \\ -7 \\ -5\end{array}\right],\left[\begin{array}{r}1 \\ -2 \\ -6\end{array}\right]$, and $\left[\begin{array}{l}3 \\ 2 \\ 3\end{array}\right]$

Answer: \left(\left[\begin{array}{l}1 \\ 1 \\ 3\end{array}\right] \times\left[\begin{array}{r}-7 \\ -2 \\ 2\end{array}\right]\right) \cdot\left[\begin{array}{r}-5 \\ -6 \\ 3\end{array}\right]=\left[\begin{array}{r}8 \\ -23 \\ 5\end{array}\right] \cdot\left[\begin{array}{r}-5 \\ -6 \\ 3\end{array}\right]=113$

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Question:
In $\mathbb{R}^{4}$ with the usual dot product, find an orthogonal basis for
$$
\operatorname{span}\left\{\left[\begin{array}{l}
1 \\
0 \\
1 \\
0
\end{array}\right],\left[\begin{array}{c}
1 \\
3 \\
1 \\
-1
\end{array}\right],\left[\begin{array}{l}
2 \\
4 \\
2 \\
2
\end{array}\right]\right\}
$$

Answer: $\mathbf{u}_{1}=\left[\begin{array}{l}1 \\ 0 \\ 1 \\ 0\end{array}\right], \mathbf{u}_{2}=\left[\begin{array}{c}0 \\ 3 \\ 0 \\ -1\end{array}\right], \mathbf{u}_{3}=\left[\begin{array}{l}0 \\ 1 \\ 0 \\ 3\end{array}\right]$

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Question:
Consider the matrix
$$
A=\left[\begin{array}{ccc}
e^{t} & \cosh t & \sinh t \\
e^{t} & \sinh t & \cosh t \\
e^{t} & \cosh t & \sinh t
\end{array}\right]
$$
Does there exist a value of t for which this matrix fails to be invertible? Explain.

Answer: Since the matrix $A$ has two identical rows, we have $\operatorname{det}(A)=0$ for all $t$. So this matrix is noninvertible for all $t$.

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Question:
Consider the inner product space $C[0,2]$, with the inner product given by
$$
\langle p, q\rangle=\int_{0}^{2} f(x) g(x) d x .
$$
Use the Gram-Schmidt procedure to find an orthogonal basis for $\operatorname{span}\left\{1, x, x^{2}\right\}$.

Answer: $\mathbf{u}_{1}=1, \mathbf{u}_{2}=x-1, x^{2}-2 x+\frac{2}{3}$.

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Question:
Decide whether
$$
\mathbf{v}=\left[\begin{array}{r}
4 \\
4 \\
-3
\end{array}\right]
$$
is a linear combination of the vectors
$$
\mathbf{u}_{1}=\left[\begin{array}{r}
3 \\
1 \\
-1
\end{array}\right] \quad \text { and } \quad \mathbf{u}_{2}=\left[\begin{array}{r}
2 \\
-2 \\
1
\end{array}\right]
$$
If yes, find the coefficients.

Answer:
$$
\left[\begin{array}{r}
4 \\
4 \\
-3
\end{array}\right]=2\left[\begin{array}{r}
3 \\
1 \\
-1
\end{array}\right]-\left[\begin{array}{r}
2 \\
-2 \\
1
\end{array}\right]
$$

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Question: Find $\operatorname{proj}_{\mathbf{v}}(\mathbf{w})$ where $\mathbf{w}=\left[\begin{array}{r}1 \\ 0 \\ -2\end{array}\right]$ and $\mathbf{v}=\left[\begin{array}{l}1 \\ 2 \\ 3\end{array}\right]$.

Answer: $\frac{\mathbf{v} \cdot \mathbf{w}}{\mathbf{v} \cdot \mathbf{v}} \mathbf{v}=\frac{-5}{14}\left[\begin{array}{l}1 \\ 2 \\ 3\end{array}\right]=\left[\begin{array}{c}-\frac{5}{14} \\ -\frac{5}{7} \\ -\frac{15}{14}\end{array}\right]$

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Question:
Let $A=\left[\begin{array}{rrr}1 & 2 & -1 \\ -1 & 4 & 0\end{array}\right]$ and $B=\left[\begin{array}{rrr}0 & 3 & 0 \\ 1 & -1 & 1\end{array}\right]$.

Find a matrix $X$ such that $(A+X)-(B+0)=B+A$. Hint: first use the properties of matrix addition to simplify the equation and solve for $X$.

Answer: The equation simplifies to $X=B+B$, so $X=\left[\begin{array}{rrr}0 & 6 & 0 \\ 2 & -2 & 2\end{array}\right]$.

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Question: Use Gauss-Jordan elimination to solve the system of equations $3 x-y-2 z=3, y-4 z=0$, and $-2 x+y=-2$.

Answer: Solution is: $[x=2 t+1, y=4 t, z=t]$

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Question:
Find $\cos \theta$ where $\theta$ is the angle between the vectors
$$
\mathbf{u}=\left[\begin{array}{r}
3 \\
-1 \\
-1
\end{array}\right], \mathbf{v}=\left[\begin{array}{l}
1 \\
4 \\
2
\end{array}\right]
$$

Answer: $\cos \theta=\frac{\left[\begin{array}{ccc}3 & -1 & -1\end{array}\right]^{T} \cdot\left[\begin{array}{lll}1 & 4 & 2\end{array}\right]^{T}}{\sqrt{9+1+1} \sqrt{1+16+4}}=\frac{-3}{\sqrt{11} \sqrt{21}}$.

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Question: Find the area of the triangle determined by the three points, $(1,0,3),(4,1,0)$ and $(-3,1,1)$.

Answer: Let $P=(1,0,3), Q=(4,1,0)$, and $R=(-3,1,1) . \overrightarrow{P Q} \times \overrightarrow{P R}=\left[\begin{array}{r}3 \\ 1 \\ -3\end{array}\right] \times\left[\begin{array}{r}-4 \\ 1 \\ -2\end{array}\right]=\left[\begin{array}{c}1 \\ 18 \\ 7\end{array}\right]$. The area of the triangle is $\frac{1}{2}\|\overrightarrow{P Q} \times \overrightarrow{P R}\|=\frac{1}{2} \sqrt{1+18^{2}+7^{2}}=\frac{1}{2} \sqrt{374}$.

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Question: Use Gauss-Jordan elimination to solve the system of equations $-9 x+15 y=66,-11 x+$ $18 y=79,-x+y=4$, and $z=3$.

Answer: Solution is: $[x=1, y=5, z=3]$

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Question:
Let
$$
A=\left[\begin{array}{llll}
3 & 2 & 0 & 0 \\
2 & 5 & 1 & 0 \\
0 & 1 & 3 & 1 \\
0 & 0 & 1 & 3
\end{array}\right]
$$
and consider the vector space $\mathbb{R}^{4}$ with the inner product given by $\langle\mathbf{v}, \mathbf{w}\rangle=\mathbf{v}^{T}$ Aw. Let
$$
\mathbf{v}_{1}=\left[\begin{array}{c}
1 \\
-1 \\
1 \\
1
\end{array}\right], \quad \mathbf{v}_{2}=\left[\begin{array}{l}
2 \\
0 \\
0 \\
2
\end{array}\right], \quad \text { and } \quad \mathbf{v}_{3}=\left[\begin{array}{l}
2 \\
0 \\
2 \\
3
\end{array}\right]
$$
and let $W=\operatorname{span}\left\{\mathbf{v}_{1}, \mathbf{v}_{2}, \mathbf{v}_{3}\right\}$. Apply the Gram-Schmidt procedure to $\mathbf{v}_{1}, \mathbf{v}_{2}, \mathbf{v}_{3}$ to find an orthogonal basis $\left\{\mathbf{u}_{1}, \mathbf{u}_{2}, \mathbf{u}_{3}\right\}$ for $W$ with respect to the above inner product.

Answer: $\mathbf{u}_{1}=\left[\begin{array}{c}1 \\ -1 \\ 1 \\ 1\end{array}\right], \mathbf{u}_{2}=\left[\begin{array}{c}1 \\ 1 \\ -1 \\ 1\end{array}\right], \mathbf{u}_{3}=\left[\begin{array}{c}-1 \\ 1 \\ 1 \\ 0\end{array}\right]$

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Question: Find the matrix for $T(\mathbf{w})=\operatorname{proj}_{\mathbf{v}}(\mathbf{w})$, where $\mathbf{v}=[1,-2,3]^{T}$.

Answer:
Recall that the desired matrix has $i^{\text {th }}$ column equal to $\operatorname{proj}_{\mathbf{u}}\left(\mathbf{e}_{i}\right)=\frac{\mathbf{u} \cdot \mathbf{e}_{i}}{\|\mathbf{u}\|^{2}} \mathbf{u}$. Therefore, the matrix is

$$
\frac{1}{14}\left[\begin{array}{rrr}
1 & -2 & 3 \\
-2 & 4 & -6 \\
3 & -6 & 9
\end{array}\right]
$$

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Question: Let $A=\left[\begin{array}{lll}-2 & 0 & 6 \\ -3 & 1 & 6 \\ -3 & 0 & 7\end{array}\right]$. Find a square root of $A$.

Answer: $A=\left[\begin{array}{rrr}0 & 0 & 2 \\ -1 & 1 & 2 \\ -1 & 0 & 3\end{array}\right]$

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Question:
Let $\mathbf{w}, \mathbf{v}$ be given vectors in $\mathbb{R}^{4}$ and define
$$
M=\left\{\mathbf{u}=\left[\begin{array}{l}
u_{1} \\
u_{2} \\
u_{3} \\
u_{4}
\end{array}\right] \in \mathbb{R}^{4} \mid \mathbf{w} \cdot \mathbf{u}=0 \text { and } \mathbf{v} \cdot \mathbf{u}=0\right\}
$$
Is $M$ a subspace of $\mathbb{R}^{4}$ ? Explain.

Answer: Yes, this is a subspace.

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Question:
Find $h$ such that
$$
\left[\begin{array}{ll|l}
2 & h & 4 \\
3 & 6 & 7
\end{array}\right]
$$
is the augmented matrix of an inconsistent system.

Answer: $h=4$.

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Question:
Find $\cos \theta$ where $\theta$ is the angle between the vectors
$$
\mathbf{u}=\left[\begin{array}{r}
1 \\
-2 \\
1
\end{array}\right], \mathbf{v}=\left[\begin{array}{r}
1 \\
2 \\
-7
\end{array}\right]
$$

Answer: $\cos \theta=\frac{-10}{\sqrt{1+4+1} \sqrt{1+4+49}}$

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Question: Find $3 \times 3$-matrices $A$ and $B$ such that $A B \neq B A$.

Answer: $A=\left[\begin{array}{lll}0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0\end{array}\right], B=\left[\begin{array}{lll}1 & 2 & 0 \\ 3 & 4 & 0 \\ 0 & 0 & 0\end{array}\right]$. Then $A B=\left[\begin{array}{lll}3 & 4 & 0 \\ 1 & 2 & 0 \\ 0 & 0 & 0\end{array}\right]$ and $B A=\left[\begin{array}{lll}2 & 1 & 0 \\ 4 & 3 & 0 \\ 0 & 0 & 0\end{array}\right]$.

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Question: What does it mean geometrically if the box product of three vectors equals zero?

Answer: It means that if you place them so that they all have their tails at the same point, the three will lie in the same plane.

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Question: Suppose the coefficient matrix of a system of $n$ equations with $n$ variables has the property that every column is a pivot column. Does it follow that the system of equations must have a solution? If so, must the solution be unique? Explain.

Answer: Yes. It has a unique solution.

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Question: Find $x$ and $y$ so that $\mathbf{u}=[5 x-3 y, 4]^{T}$ and $\mathbf{v}=[2 x-2 y, 2 y]^{T}$ are equal in $\mathbb{R}^{2}$.

Answer: We need $5 x-3 y=2 x-2 y$ and $4=2 y$. The unique solution is $x=\frac{2}{3}$ and $y=2$.

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Question: Let $V$ be a 5-dimensional vector space. If you have 5 linearly independent vectors in $V$, can you conclude that the vectors span $V$ ?

Answer: Yes, because the set of 5 linearly independent vectors can be extended to a basis $B$ of $V$. But since $V$ is 5-dimensional, $B$ has only 5 elements, which must be the original 5 vectors.

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Question:
Consider $\mathbb{R}^{3}$ with the usual dot product. Let
$$
\mathbf{u}_{1}=\left[\begin{array}{r}
-1 \\
1 \\
1
\end{array}\right], \quad \mathbf{u}_{2}=\left[\begin{array}{r}
-1 \\
-2 \\
1
\end{array}\right], \quad \text { and } \quad \mathbf{v}=\left[\begin{array}{r}
-1 \\
5 \\
3
\end{array}\right]
$$
Note that $\mathbf{u}_{1}$ and $\mathbf{u}_{2}$ are orthogonal. Find the best approximation of $\mathbf{v}$ in $\operatorname{span}\left\{\mathbf{u}_{1}, \mathbf{u}_{2}\right\}$.

Answer: The best approximation is $\mathbf{v}^{\prime}=3 \mathbf{u}_{1}-\mathbf{u}_{2}=\left[\begin{array}{r}-2 \\ 5 \\ 2\end{array}\right]$.

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Question:
Solve the following system of equations by back substitution.
$$
\begin{array}{r}
x+3 y-2 z=5 \\
y+3 z=4 \\
z=1 .
\end{array}
$$

Answer: $(x, y, z)=(4,1,1)$.

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Question:
Find $h$ such that
$$
\left[\begin{array}{ll|r}
1 & 1 & 4 \\
3 & h & 12
\end{array}\right]
$$
is the augmented matrix of a consistent system.

Answer: Any $h$ will work.

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Question: Use Gauss-Jordan elimination to solve the system of equations $-19 x+8 y=-108,-71 x+$ $30 y=-404,-2 x+y=-12,4 x+z=14$.

Answer: Solution is: $[x=4, y=-4, z=-2]$

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