by Judith Curry
I spotted this presentation by Arthur Dempster, Harvard statistician, in the Series on Mathematical and Statistical Approaches to Climate Modeling hosted by the Isaac Newton Institute for Mathematical Sciences.
Dempster is widely known as the co-originator of Dempster-Shafer Evidence Theory (see the Wikipedia for an overview). Elements of evidence theory have been discussed on several previous threads (see Italian Flag, reasoning about floods).
I find this presentation to be quite provocative. Here are some excerpts:
One concern is the basic problem of trying to get physicists and statisticians on the same page. Statisticians think of themselves as dealing with data as “information with context”. . . and with statistical models as parts of a complex system of tools for extracting meaning from statistical data. Physicists on the other hand tend to think of models as approximations to scientific truth, with the ultimate goal of research being to arrive at representations and explanations of such truth. The two cultures are very different.
Almost in contradiction to their pure science backgrounds, it has become a basic function of physical climate modelers to inform policymakers and other real world stakeholders about possible alternative future climates. When used in this mode, climate models are treated as carriers of information, so move closer to statistical models. Specifically, physical models become interpretable as information when their equations are regarded as approximating relations among the values of actual real world variables at successive points in actual time. Statistical models, should similarly be regarded as describing probabilistic relations among unknown true values of such variables, including probabilistic time dependence.
Approaching Probabilistic Models: What are the Issues?
What are the problems and prospects for moving from weather to the longer time scales of climate change? What unknowns are predictable probabilistically on longer time scales, and which are not?
A fundamental issue concerns the nature of uncertainty. All can agree that predictions are uncertain. But what mathematics should be used when computing and reporting predictive uncertainties? Here the divergence of the two cultures is astonishing. Few physicists have training and hence knowledge of how the mathematics of probability, together with its relations to scientific uncertainty, has developed over 300 years into a formidable set of theoretical structures and tools. The identity of the academic discipline of statistics was transformed, especially over the middle decades of the 20th Century, by competing methodologies for addressing scientific uncertainty. This is not the place to delve into explaining what developed, and how there are differing viewpoints, with mine in particular lying outside the statistical mainstream.
I believe it is fair to say, however, that how physicists approach scientific uncertainty has been scarcely touched by fundamental developments within statistics concerning mathematical representations of scientific uncertainty. An indication of the disconnect is provided by the guidelines used by the IPCC in its 2007 major report, where the terms “likelihood” and “confidence” were recommended for two types of uncertainty reports, apparently in complete ignorance of how these terms have been used for more than 60 years as basic textbook concepts in statistics, having nothing whatsoever in common with the recommended IPCC language (which I regard as operationally very confusing). Another indication is that experts from the statistical research community constituted according to one source only about 1% of the attendees at the recent Edinburgh conference on statistical climatology.
Roles for Statistical Modeling
Physical modelers often refer to two basic sources of uncertainty when interpreting the output of a climate simulator, namely, uncertainty about initial conditions, and uncertainty from discretization, or transform truncation, of space/time variables. From my outsider’s perspective, I would prefer an emphasis on attempting to model and analyze only the unique actual climate system, instead of the current practice of running and analyzing a series of mathematical and therefore artificial climate systems. Of course, the same pair of uncertainty sources arise in the combined physical/ statistical modeling approach that I am advocating.
My suggested model type is captured by the term “hidden Markov model”. The thing that is hidden is the actual past, present, and future of the real climate system, which is the domain of physical thinking and modeling that proceeds forward in time, such as may be represented, for example, by the equations of AOGCMs. Since the real processes are hidden, they cannot be directly simulated. Alongside the hidden system there is empirical data also linked to real space/time, and partially obscured from the actual system by observational error. The goal of hidden Markov analysis is to update posterior probability assessments of the true system, including limited ranges of past, present, and future, given stepwise accrual of empirical data. It is these probability assessments that should be updated sequentially. Once models are specified, this becomes a defined computational task for Bayesian or DS analysis. Fast algorithms are known. They may look like simulations from a physical model, but are conceptually different because they sample the posterior probability assessment from fused empirical and theoretical information sources, typically using MCMC methodologies.
JC comment: hidden Markov models is something new to me, here is the Wikipedia description.
One task for the statistical research community is to formulate and implement probabilistic space/time models, from which principles of statistical inference determine posterior probabilistic assessments of the true climate, to whatever level of detail the assumed state space permits. For the past up to the present, hidden Markov analyses, such as the familiar Kalman filter, or more complex versions thereof, fuse the information from the past concerning the actual process with information from the current empirical record. For predicting the future climate, there is no data, so statistical error models are no longer operational. The climate proceeds on its own with probabilistic uncertainty entering only through probabilistic uncertainty about the present state of the actual system. Predictive analysis then proceeds by forward propagation of probabilistic uncertainty.
The necessary discreteness of physical models suggests that they might best be regarded as tracking local averages across neighboring space/time regions. Because they are approximations to relations believed to hold for infinitesimal changes across time and space, they reflect model errors arising from the inability to represent natural processes in space/time domains smaller than discretization can capture. Physical modelers typically introduce “parameters” that attempt to adjust difference equations for the missing processes. I sense that statisticians could become more deeply involved in the development of probabilistic representations of such discretization errors, which in effect turn even the non-empirical component of the model into a parametrized stochastic process whose parameters need to be assessed through formal statistical inference tools (e.g., Bayes or DS).
The Problem of Chaos
Presumably most detailed physical climate models of the atmosphere predict chaotic instabilities in the real world climate system, analogous to those that make longer range weather forecasting a low skill enterprise. How might these instabilities impact the task of devising credible probabilistic predictions of long term trends in future climates? My response is to make a radical proposal, linking the recognized difficulty of predicting chaotic systems as the future time horizon grows with a fundamental change in how probabilities should be similarly degraded on a similar time scale.
The proposal is based on a weakening of Bayesian theory that I originally developed in a series of papers in the 1960s. Further developments were spearheaded by Glenn Shafer in the 1970s and 1980s ,who gave the theory an AI spin, and named it the theory of belief functions. I now prefer to call the DS (for Dempster-Shafer) calculus. It appears to be gradually gaining increased recognition and respect.A detailed exposition of DS is not possible in this note, but I wish to draw attention to two basic features of the DS system. The first is that probabilities are no longer additive. By this I mean that if p denotes probability “for” the truth of a particular assertion, here some statement about a specific aspect of the Earth’s climate in the future under assumed forcing, while q denotes probability “against” the truth of the assertion, there is no longer a requirement that p + q = 1. Instead these probabilities are allowed to be subadditive, meaning thatingeneral p+q<1.Thedifference1-p-qislabeledr,sothat now p + q + r = 1, with r referred to as the probability of “don’t know”. (Note: each of p, q, and r is limited to the closed interval [0,1].)
It is only for the last two years that I have focused on trying to explain what is meant by the DS concept of “don’t know”. I was helped when I ran across a reference to the following remarks by economist John Maynard Keynes, remarks that I believe have not been taken sufficiently seriously:
By “uncertain” knowledge, let me explain, I do not mean merely to distinguish what is known for certain from what is only probable. The game of roulette is not subject, in this sense, to uncertainty; nor is the prospect of a Victory bond being drawn. Or, again, the expectation of life is only slightly uncertain. Even the weather is only moderately uncertain. The sense in which I am using the term is that in which the prospect of a European war is uncertain, or the price of copper and the rate of interest twenty years hence, or the obsolescence of a new invention, or the position of private wealth-owners in the social system in 1970. About these matters there is no scientific basis on which to form any calculable probability whatever. We simply do not know. Nevertheless, the necessity for action and for decision compels us as practical men to do our best to overlook this awkward fact and to behave exactly as we should if we had behind us a good Benthamite calculation of a series of prospective advantages and disadvantages, each multiplied by its appropriate probability, waiting to he summed. (Excerpted from “The General Theory of Employment”, Quarterly Journal of Economics, February, 1937, pages 209-223.)
A second fundamental feature of the DS calculus is a particular “rule of combination”, or principle, for combining information from different sources, such as physical models and empirical data concerning past and present. The DS rule is linked to a DS concept of independence, which might therefore be regarded as a severe restriction on its applicability, except that many special cases of the rule are routinely used with little overt concern about independence, including Bayesian combination of likelihood and prior, and Boolean logical combination. Independence in the mathematics of ordinary additive probabilities is another special case. Most models in the burgeoning field of applied probability can be viewed as constructed from many independent components.
The DS rule of combination is a powerfully inclusive tool of probabilistic analysis, with potentially important applications to probabilistic climate prediction. In particular, I hypothesize that DS-style probabilities of “don’t know” could come to be a basic way to separate unpredictable from predictable aspects of climate change. From the DS perspective, Bayesian inference cannot do this in a satisfactory way. An simple illustration may help to support my argument. It is easy to find on the web beautiful discussions and simulations of simple chaotic systems, beginning from the simple 3D model popularized by the late Ed Lorenz. The basic problem is that small perturbations in initial conditions often grow into large perturbations over “time”. A Bayesian approach puts a prior on the initial position, which may be tiny, yet soon is projected forward to a much more spread out marginal distribution. The result is typically a limiting predictive distribution over the whole system. In a weaker DS framework, the prior may simply be a small region about initial point, where you “don’t know” where the true initial condition is in the small region, i.e., you have r = 1 for the (p, q, r) of that region. As time progresses, your predicted “don’t know” region with r = 1 grows, possibly taking over the whole system. It looks to me pretty obvious that the DS option of a logical (i.e., nonprobabilistic) analysis is needed to represent the fade-out of predictability for chaotic systems. Additive Bayesian predictive posterior distributions are unable to function in this way.
The Problem of “Complex Systems”
Climate prediction not only has the problem of nonlinearity in dynamical systems, but also shares with the analysis of typical real complex systems the equally great trouble associated with the presence of a huge panoply of variables, subsystems, and possible feedbacks in play. I remember Rol Madden commenting, back in the 90s when I used to visit NCAR, that it would be a sheer accident if numerical experiments with GCMs were to give credible quantitative representations of the real climate, presumably including future effects of increasing GHG concentrations.
Suppose that the system was defined at the outset to include the full carbon and hydrological systems. Then add atmospheric and ocean chemistry, and then living and breathing systems everywhere. Much “don’t know” abounds simply about the present and recent past, let alone the fundamental problem of getting quantitative about the future.
What does this say about research priorities? Given the real climate system, characterized not only by fundamental “don’t know” coming from dynamical nonlinearities, but also fundamental “don’t know” coming simply from an inability to supply meaningful evidence-based priors in the presence of complexity, I believe that communities faced with needs for real predictions of complex systems should be investing in models of probabilistic prediction that provide measures of the type of “don’t know” described by Keynes, including the DS approach as a leading candidate for numerical implementations.
JC comments: there are some powerful and new ideas here of relevance to climate modeling, notably the formal inclusion of “I don’t know”. I don’t quite understand all of this or how it might work in the context of climate modeling, I look forward to your interpretations and discussion.


Recent Tutoring Session Reviews
"I completed a test review study guide with the student. We reviewed many problems involving areas and perimeters. Some basic trig functions were also utilized. She will do well on her test."
"We reviewed for the student's chapter 8 exam. We went over the important formulas that she would have to be familiar with in order to answer questions about surface area, lateral area, volume, arc length, sector area, and the segment of a circle. She seemed capable of using the appropriate formula for each question, but would occasionally stumble and incorrectly apply a formula. We will work on that in future sessions. Overall, however, she seemed to have a good grasp of the material."
"The student has an exam coming up. While last session we reviewed the assigned practice packet, this session we reviewed practice problems that he selected from the book. He seems confident with most of the material that will be presented on the exam. The questions that he had trouble solving at first had an algebraic component to them, and he was able to figure out how to solve them upon review. The majority of this session focused on test-taking strategy, such as not investing too much time on one problem the first time around. It seems like the student is able to work through more difficult problems by skipping them initially and going back to them later."
"We answered the student's homework about circles. He finished all of the questions. He was refreshed on circles theorem and calculations."
"The student had another very good session. We are working our way through multiple geometric concepts. Today the focus was on right triangles, area and perimeter, and coordinate geometry. Once he learned the concepts he had no problems. We will meet again on Saturday."
"The student completed his weekly packet of practice problems. I was happy to find that he had begun the packet and gotten every problem that he tried correct. After completing the packet we worked on problems from his Geometry book to prepare for his finale. He demonstrated that he could properly identify when to use the slope formula and use it correctly. He did the same with the distance formula. I was very happy to see he could now do this without any help. He also has a strong understanding of two-dimensional shapes, which is very important. We will be reviewing for his final for our remaining sessions. He has a good attitude, as always. He seems very pleased with his progress and is gaining confidence, which is great."


The Ultimate Showdown
The goal of this study is to find investigate Olympic discus throwers at their peak performance, and to find out if and when women's performance will pass men's performance, or vice versa. This is done with data from olympic.org, and calculated using linear regression.
Data Table 1
This table shows the data of gold medalists from 1972-1984. The data is shown in meters (how far the discus is thrown).
Data Table 2
Data Table 1 continued. Contains data from 1988-2000.
Data Table 3
The final table and a continuation of Data Table 2. Contains data from 2004-2012.
The Turning Point
The women were originally dominating the men in discus, excelling their scores by a fair margin. Though as time passed, women's performance declined, and conversely, men's performance increased. The intersection point of the two lines is the point where men's performance surpasses women's performance, in 1997. Since that is not an Olympic year, men have been outperforming discus since the 2000 Olympics in Sydney, Australia.
The X Coordinate
This coordinate on the graphs above denote the year that each competition occurs. This is shown in relation to the y coordinate.
The Y Coordinate
This coordinate on the graphs above describe the distance that the competitors threw the discus in meters. This can produce data when combined with the figures from the x coordinate.
"Senior Presentation Discus | Publish with Glogster!" Glogster. N.p., n.d. Web. 04 Feb. 2015.
"Who Makes A Better Leader: A Man Or A Woman?" Forbes. Forbes Magazine, n.d. Web. 04 Feb. 2015.


I’m confident that early conversations concerning Math lead to deeper understanding with Mathematical concepts. It’s appropriate to say to a 3-year-old “How many cookies do you have”? (considering the number is less than five).
While making sandwiches, I might say. “I am making three sandwiches”. Another thing a parent might say, “If I have three apples then, I eat one …. How many apples will I have left?”
Math is a concept that builds. In my experience, a strong foundation in adding, subtracting, multiplying and dividing usually in that order leads to a lifetime of ease in Mathematical concepts.
Simply having your child name then count his favorite friends at school not only is a great conversation piece yet, also allows the parent to grasp the conceptual knowledge of the child’s mathematical understanding.
This should definitely be age appropriate. A four year with 4 stickers can be expected to count up to the number 4 or 5 with practice. The idea of being able to count simple objects is another skill that comes with practice. For example, if the dad has 3 legos in his hand. Have your child point to each lego counting slowly while pointing to each lego. One to one correspondence is a skill your little one might need a little more practice with; however, with patience and care this skill will be mastered.
These early Math conversations as well, as reading books that highlight Math concepts, such a simple counting books — leads to a lifetime of learning.
There are so many learning activities that surround play dough. Play dough activities offer a wonderful age appropriate opportunity for learning. Easy play dough recipes can be found online using simple household ingredients.
Simple counting videos and activities is another great way to encourage mathematical concepts.
Have Fun With Math,


(note: I think OP's issue has more to do with confusing the concepts behind probability theory and random variables than with what "i.i.d." really means. So I'll try to give an informal explanation. For the math of i.i.d., @Taylor's response is quite on point.)
Random variables are abstractions that you use to represent non-determinstic problems, not coins, dice, or other "random objects". So there are no strict rules saying that one coin toss is $X$, two coin tosses are $X_1$ and $X_2$, etc.
The interesting thing about modeling a coin as one single variable $X$ is that it creates an abstraction for all coins, or all possible realizations of a coin tosse. As long as your coin is fair, any time you toss it, you can think of a variable $X$ that follows a Bernoulli distribution.
If you are studying a succession of coin tosses, you could come up with the most contrived ways of modeling it. For instance, you could think of a random variable $Y$ that represents all possible outcomes of a sequence of $n$ coin tosses. Or you could do something even weirder, such as a random variable $Y_3$ for the combined outcomes of the first 3 tosses, and another random variable $Y_r$ for the combined outcomes of the remaining tosses.
But that would be silly. Why? Because there are two interesting things about tossing a coin $n$ times:
- each toss does not affect the others (independence);
- all tosses are identical -- the chance of heads coming up in each toss is always 0.5 (identical distributions).
So, to answer your third question:
Or are $X_1$, $X_2$,..., $X_n$ all considered as random variables if we want to consider them IID?
You don't "consider" them random variables. There are no such things as $X_1$, $X_2$,..., $X_n$ in the real world and nothing saying that you must think of $X_1$, $X_2$,..., $X_n$ when you think of $n$ coin tosses.
However, when you think of those $n$ variables and their probability distributions, you get to say that they are all i.i.d. because of properties 1 and 2 above. And in probability theory there's all sorts of useful math that other people discovered you can use when you have a collection of i.i.d. variables.
To make it clear, these variables are independent because they represent independent events (each $X_i$ is related to a different, independent toss). And they are identically distributed because they all follow the same distribution: the Bernoulli probability distribution with $p=0.5$.
So what happened to $X$? Nothing. Notice that you invented those random variables $X_1$, $X_2$,..., $X_n$ because tossing a coin $n$ times is a different problem than tossing a coin a single time. The interesting thing about $X$ is that each variable $X_i$ has, individually, the same properties as your original variable $X$.
Now back to your remaining questions:
If $X$ denote a random variable which means the "result of a coin toss" then $x_1$, $x_2$, ..., $x_n$ are the results of repeated coin tossing. Are $x_1$, $x_2$, ..., $x_n$ IID? If yes, how can observations be variables? [...]
How can a random sample be IID?
The terminology in probability theory can be complicated sometimes, because the concepts are all so closely related. But here's what I (think) I know.
Strictly speaking, the correct would be to say that $x_1$, $x_2$, ..., $x_n$ are drawn from a collection of i.i.d. variables $X_1$, $X_2$, ..., $X_n$.
However, even though the observations only exist after the coin has been tossed (and therefore it's not random anymore), we are still speaking in general terms. We are talking about any possible situation where you could have those $n$ observations, not about a particular set of observations that you know because you just tossed a coin many times. So we don't know them. We might just as well treat them as random variables.
Therefore, even though observations are not random variables, it makes sense to associate with them many properties that actually belong to random variables. By extension we can thus say that $x_1$, $x_2$, ..., $x_n$ are i.i.d. because they were all drawn independently and from identical distributions.


Tycho plans his running training. Each week he wants to go for a run on the same weekdays. He never wants to go for a run on two consecutive days. But he wants to go for a run two days a week. How many different weekly plans meet those conditions?
There are 18 possible weekly plans.


First of all we worked out how to make each digit from 0 to 9 as if it were on a calculator. We found out that the number 8 uses the most ice-block sticks, you need 7, and the number 7 and 9 have extra sticks that we didn't expect them to have.
Then we had to get 10 ice-block sticks each. We checked we had the right number by arranging our sticks into a tally chart.
Then the challenge was issued.
Make the biggest number you can out of 10 ice-block sticks.
We learnt that to make a really big number you need to make as many digits as you can. Our biggest number was a 5-digit number. It contained all 1s. It was eleven thousand, one hundred and eleven. It was even bigger than four-digit numbers we made with a 7 in them.


I've been looking at an erasure code by Luigi Rizzo which is based on Vandermonde matrices, since this code is used in Outernet. In fact, it is the code implemented by the zfec library. Luigi Rizzo describes his code in a paper from 1997, but the paper can be very confusing and misleading because it describes the mathematics in very little detail. I needed to go to the source code to understand how it works. Actually, the idea behind this code is very simple. Here I do a mathematical description of the code and show that it is the same as a Reed-Solomon code. This is rather weird, because Luigi Rizzo makes no mention of Reed-Solomon codes, which were first described in 1960.
Let be a field. Recall that an linear code over can be given in terms of a generator matrix , which is an matrix with entries in and rank . A message is encoded as , where is computed as (here I always consider vectors as columns). The code is called systematic if has the form
so that for . Note that this is really a property of the matrix , since the code is defined just as the image of .
We are interested in designing an systematic code that works as an erasure code and is able to correct up to erasures. This means that we should be able to compute from even if only of the components of are known. This condition is equivalent to the condition that every submatrix of is invertible. To see this, note that if we regard the equation as a linear system for and consider only the equations for which we know the value of , then we obtain a reduced linear system , where is the vector of the components of which we know and is the correspondent submatrix of . Therefore, needs to be invertible so that we can find in terms of .
An idea to find such a matrix is to use Vandermonde matrices. If , the Vandermonde matrix is
Its determinant is
so the Vandermonde matrix is invertible if and only if the elements are distinct.
The matrix is constructed in the following way: we choose distinct elements and we put
write and define
so that has the required form to give a systematic code. This matrix satisfies the condition that every submatrix of is invertible, because such a submatrix is of the form , where is the corresponding submatrix of . Note that is a Vandermonde matrix constructed with distinct elements, and hence invertible, so is also invertible. In the intended application, is the finite field with elements, so up to distinct elements can be chosen, and can be at most .
The keen reader will have noticed that this construction looks very much like a Reed-Solomon code. In fact, the code described by Luigi Rizzo is the same as a Reed-Solomon code. The only difference is that Luigi Rizzo is only concerned with recovering erasures, which is much easier than recovering errors. The fact that Luigi Rizzo doesn't mention Reed-Solomon codes makes me suspect he didn't have a good knowledge of coding theory. The authors of zfec seem more knowledgeable and they mention that zfec is a Reed-Solomon code, albeit only in some of the comments in the source code. This doesn't seem to be widespread knowledge, so I'll say it again clearly:
zfec is just a Reed-Solomon code used as an erasure code
To see that this code is the same as a Reed-Solomon code, let us recall the systemmatic encoding of Reed-Solomon codes. As above, is the finite field with elements. To construct an Reed-Solomon code over , we choose distinct elements . Given , there is a unique polynomial of degree less than such that for . Indeed, , where the vector is obtained as . Here denotes the Vandermonde matrix as above. The message is encoded to produce , where for . Thus, , where and are defined as above. We see that the systematic generator matrix for this Reed-Solomon code is the same as the generator matrix considered by Luigi Rizzo.


Schemes of Work
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Highest Common Factor and Lowest Common Multiple
4 part lesson on finding the highest common factor and lowest common multiple of two or more numbers.
Indices with Fractional Powers
4 part lesson on simplifying numbers with negative and fractional powers.
Adding Fractions with Different Denominators
4 part lesson on adding fractions with different denominators.
4 part lesson on converting recurring decimals to fractions.
4 part lesson on multiplying with fractions and mixed numbers.
Mixed Numbers and Top Heavy Fractions
4 part lesson on converting between mixed numbers and top-heavy fractions.
Calculating a Fraction of an Amount
4 part lesson on calculating a fraction of an amount.
4 part lesson on calculating equivalent fractions.
Dividing Fractions and Mixed Numbers
4 part lesson on dividing with fractions and mixed numbers.
4 part lesson on using equivalent fractions to perform long division.
4 part lesson on multiplication with decimal numbers using the grid method.
Ordering Decimal Numbers
4 part lesson on ordering decimal numbers.
Multiplying and Dividing by Tens
4 part lesson on multiplying and dividing by 10, 100 and 1000.
Volume of a Cone
4 part lesson on calculating the volume of a cone.
Rounding to a Decimal Place
4 part lesson on rounding numbers to a given decimal place.
Rounding to Significant Figures
4 part lesson on rounding numbers to a significant figure.
Mr Mathematics Blog
Trigonometric Identities Sin, Cos and Tan
September 15, 2020
How to introduce the sin, cos and tan trigonometric identities.
Calculating a Reverse Percentage
September 3, 2020
How to teach calculating the original amount after a percentage change.
Comparing Datasets using the Mean and Range
August 25, 2020
The importance of the range when comparing comparing datasets.


I’ll Savor the Chocolate, Unplug the Drama
Happy Scud Day! Scud stands for Savor the Comic, Unplug the Drama. Relax and enjoy the funnier things in life! It’s also a day to celebrate mathematics, as it’s Math 2 point o Day. Do some math equations just for fun! Finally, have some chocolate and almonds for Chocolate With Almonds Day!


Did you know spinning a prize wheel can involve math? Give your child some practice with probability, and he'll determine how likely each outcome of the prize wheel is. This will help him understand the concept of fractions and statistics. Download Free Worksheet. See in a set (9) View answers. Add to collection. Assign digitally. Grade..
Aumentar a chance de drop dos itens equipáveis em 250%.
Fazer com que uma morte conte como x3 no Bosstiary.
Probability distributions calculator. Enter a probability distribution table and this calculator will find the mean, standard deviation and variance. The calculator will generate a step by step explanation along with the graphic representation of the data sets and regression line.
Prize WheelProbability. Answer the probability questions related to the prize wheel and the letter cube. 4th through 7th Grades. View PDF. Probability Cards (Intermediate) What is the probability of choosing a particular card from a deck? Requires basic knowledge of standard playing cards.
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A lua está cheia! Cuidado, criaturas licantrópicas como werewolves, werefoxes ou werebears vagam pelas terras agora. E eles são mais agressivos e numerosos do que o normal. dynamics 365 trial environment
The probability P (stun) of one or more stuns per attack is equal to 1-P (no stun). For a single bullet, the probability of no stun is (1-0.5*0.0125) assuming a hit chance of 50%. Therefore the probability of at least one bullet giving a stun is 1- (1-0.5*0.00125)^40 = 0.0246 or roughly 2.46% per attack.
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Probability Distributions for SimulationFor experienced modelers, the most challenging task in creating a simulation model is usually not identifying the key inputs and outputs, but selecting an appropriate probability distribution and parameters to model the uncertainty of each input variable. For example, Risk Solver software provides over 40 probability distributions -- so which one.
Single Lottery Ticket Function: Now let's calculate the probability of winning with a single ticket purchased. def one_ticket_probability(numbers): total_outcomes = combinations (49, 6) successful_outcome = 1 / total_outcomes. probability_winning = (successful_outcome / total_outcomes) *100.
Let’s look at a European Roulettewheel. Over a single spin, the odds of a red number coming up are 18/37, or 1 in 2.06. Over 2 spins, the odds of 2 reds coming in are 1 in 2.06×2.06, or 4.24. The odds of seeing 10 reds in succession are 1 in (2.06) to the power of 10 = 1 in 1376.
You play a game that involves spinning the money wheel shown. You spin the wheel twice. Find the probability that you get more than $500 on your first spin and then go bankrupt on your second spin. Answer: Question 12. PROBLEM SOLVING You play a game that involves drawing two numbers from a hat.
Lottery Wheel and Number Patterns in Unison. The lottery wheel is all about "combinatorics." The combinatorial pattern is all about "probability." In mathematics, these two concepts are separate fields because each concept serves a different application in real life. So we shouldn't say "probability versus combinatorics."
Play A Spinning Wheel For Cash Prizes At Funzpoints Get a free spin every four hours at Funzpoints Casino for a chance to win a piece of the $1,000 daily prize pool. Funzpoints is a sweepstakes casino that accepts players in almost all US states, including California, New York, Texas, and Florida.
The players can bet on any number from 0 to 36. For example, let's say that Kriti bets £100 on the number 5. Now, a ball will be dropped into the wheel, which is then given a spin. If the ball lands on the pocket marked 5, Kriti will win (£100) * 36 = £3,600, resulting in net winnings of £3,600 - £100 = £3,500. However, if the ball ...


For a fire is kindled in My anger, And shall burn to the lowest hell; It shall consume the earth with her increase, And set on fire the foundations of the mountains.(Deuteronomy 32:22) NKJV
It’s very useful to know the volume of material in the Earth’s crust and mantle, but before we move on it’s also worthwhile to note that the Earth’s crust and mantle have several Biblical synonyms.
Crust and Mantle, Synonyms
|bottoms of the mountains||לְקִצְבֵ֤י הָרִים֙||qetseb har||Jonah 2:6|
|foundations of the hills||וּמוֹסְדֵ֣י הָרִ֣ים||mowcadah har||Psalms 18:7|
|foundations of the mountains||וּמוֹסְדֵ֣י הָרִ֣ים||mowcadah har||Deut. 32:22|
|foundations of the world||מוֹסְד֪וֹת תֵּ֫בֵ֥ל||mowcadah tebel||Psalms 18:15|
This isn’t a particularly complex problem. We can estimate the volume of material that’s in the earth’s mantle at the present time by using the boundaries of the regions of the earth that we get from seismological data.
If we assume that this volume of material was a hollow sphere, of which the inner radius is the same as the present day earth surface, then we can calculate the radius of the earth on the 3rd day. There is a useful tool that helps with these calculations which may be found here.
Earth, Present Day (values)
|Earth, Present Day||Approx. radius (km)||v||Approx. volume (m3)|
|Inner surface of mantle||4,100||v2||2.89E+11|
|Mantle material (v1-v2)||v3||7.95E+11|
|3rd day total volume (v1+v3)||7,653||v4||1.88E+12|
Volume of Earth’s Mantle
|Earth total volume||p||6,371 km||1.08E+12 m3|
|Volume of core and great gulf||g||4,100 km||2.89E+11 m3|
|Volume of mantle material||p-g (m)||7.95E+11 m3|
The total volume of present day Earth is 1.08E+12 m3
The volume of material in the crust and mantle is found by subtracting the volume of a sphere of radius 4,100 km from the volume of a sphere of radius 6,371 km
The area between the lower mantle and the core is open space, the great gulf. There’s no inner and outer core. That’s just a contrivance of popular science (SciPop) to make it seem like they know what they’re talking about, and they want to believe (and they want you to believe) that there’s no hell. The core is hell. This isn’t a euphemism.


CAN MONEY BUY SUCCESS?
We all know money can't buy love or happiness. In professional sports, can spending more money on players buy a team more success? Let's analyze this question for the NFL, NBA, and MLB.
Does a Larger Payroll Buy Success in the NFL?
In chapter 40 we learned how to calculate offensive and defensive power ratings for NFL teams. For example, an offensive team rating of + 3 means a team (after adjusting for the strength of opposition) scores 3 points more than average and a defensive rating of −5 means that (after adjusting for the strength of opposition) that a team gives up 5 fewer points than average. For the 2001–4 seasons we tabulated the amount of money each NFL team paid their offensive and defensive personnel. ...


A subject of growing interest in mathematics education is the affective domain and its effects on the teaching and learning processes, giving rise to different models of its components and conditioning factors. In this paper, we apply the ontological and semiotic categories from the Onto-Semiotic Approach (OSA) to research in mathematics education, to build an inclusive and systematic model to consider affective situations, practices, objects and processes, as well as the corresponding dualities: personal – institutional, ostensive – non-ostensive, extensive – intensive, unitary – systemic, expression – content. The dynamic character of affects (emotions, attitudes, beliefs and values) and their relations with the epistemic, cognitive, interactional and resources is modelled by the didactical configuration and didactical trajectory notions, theoretical tools which include the affective sub-configuration and sub-trajectory as key components. Another result obtained from this work is the revision of the indicators of affective suitability proposed in previous works.
Beltrán-Pellicer, P., & Godino, J. D. (2020). An onto-semiotic approach to the analysis of the affective domain in mathematics education. Cambridge Journal of Education, 50(1), 1-20. DOI: 10.1080/0305764X.2019.1623175


John Smolin coined the phrase "Going to the Church of the Larger Hilbert Space" for the dilation constructions of channels and states, which not only provide a neat characterization of the set of permissible quantum operations but are also a most useful tool in quantum information science.
According to Stinespring's dilation theorem, every completely positive and trace-preserving map, or channel, can be built from the basic operations of (1) tensoring with a second system in a specified state, (2) unitary transformation, and (3) reduction to a subsystem. Thus, any quantum operation can be thought of as arising from a unitary evolution on a larger (dilated) system. The auxiliary system to which one has to couple the given one is usually called the ancilla of the channel. Stinespring's representation comes with a bound on the dimension of the ancilla system, and is unique up to unitary equivalence.
Stinespring's dilation theorem
We present Stinespring's theorem in a version adapted to completely positive and trace-preserving maps between finite-dimensional quantum systems. For simplicity, we assume that the input and output systems coincide. The theorem applies more generally to completely positive (not necessarily trace-preserving) maps between C * − algebras.
- Stinespring's dilation: Let be a completely positive and trace-preserving map between states on a finite-dimensional Hilbert space H. Then there exists a Hilbert space and a unitary operation U on such that
- for all , where denotes the partial trace on the system.
- The ancilla space can be chosen such that . This representation is unique up to unitary equivalence.
It is sometimes useful not to go to a larger Hilbert space, but to work with operators between the input and output Hilbert spaces of the channel itself. Such a representation can be immediately obtained from Stinespring's theorem: We introduce a basis of the ancilla space and define the Kraus operators tk in terms of Stinespring's unitary U as
The Stinespring representation then becomes the operator-sum decomposition or Kraus decomposition of the quantum channel T:
- Kraus decomposition: Every completely positive and trace-preserving map can be given the form
- for all . The Kraus operators satisfy the completeness relation .
Purification of quantum states
where is a pure state on the combined system . In other words, every mixed state can be thought of as arising from a pure state on a larger Hilbert space. This special version of Stinespring's theorem is usually called the GNS construction of quantum states, after Gelfand and Naimark, and Segal.
For a given mixed state with spectral decomposition , such a purification is given by the state
References and further reading
- M. A. Nielsen, I. L. Chuang: Quantum Computation and Quantum Information; Cambridge University Press, Cambridge 2000
- K. Kraus: States, Effects, and Operations; Springer, Berlin 1983
- E. B. Davies: Quantum Theory of Open Systems; Academic Press, London 1976
- V. Paulsen: Completely Bounded Maps and Operator Algebras; Cambridge University Press, Cambridge 2002
- M. Keyl: Fundamentals of Quantum Information Theory; Phys. Rep. 369 (2002) 431-548; quant-ph/0202122
- W. F. Stinespring: Positive Functions on C * − algebras; Proc. Amer. Math. Soc. 6 (1955) 211
- I. M. Gelfand, M. A. Naimark: On the Imbedding of Normed Rings into the Ring of Operators in Hilbert space; Mat. Sb. 12 (1943) 197
- I. E. Segal: Irreducible Representations of Operator Algebras; Bull. Math. Soc. 61 (1947) 69


Hi, I've been looking for some of the early work done on transitting theory for the detection of extrasolar planets. I am especially interested in the calculation of the transitting time. The only equation I have so far come across is... T/P=(1/pi)*arcsin(R*/a) in the paper by Rivera et al 2005 on the detection of GJ 876d. It strikes me that this equation (although not formally stated in the paper) is only an approximate (and was unreferenced so I can't trace where it came from). If anyone knows of a more exact equation and some work which has been done on this subject, please post a reference. Thanks.


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The Counting 2 exercise appears under the ***. This exercise used to be on one of the missions under The World of Math, but it was removed due to reordering, renaming, or other changes. This exercise plays with some of the counting techniques.
Types of Problems
There are three types of problems in this exercise:
- Combination: This problem has a selection from a group of objects where order is not relevant. The student is asked to fid out how many possibilities there are in the situation.
- Multiplication principle: This problem has a situation described where multiple selection are needed. The student is asked to find the number of total possibilities.
- Arithmetic sequence: This problem describes a person counting from a certain number by a certain amount. The student is asked to find the number that the counter will stop on.
This exercise is easy to get accuracy badges and speed badges because although the numbers change, the problems themselves are very consistent in their difficulty. User just needs to learn how to do the three types.
- The Combination problem seems to always be handshakes. If it is, a combination where user chooses two can be performed by taking the given number of people, subtracting one, multiplying those numbers together, and dividing by two.
- The multiplication principle means to take the possible outcomes for each part of the event, and multiply those numbers together for the total number of possibilities.
- For the Arithmetic sequence problem user can use arithmetic sequences from pre-calculus. In other words, take the first number and add the product of the difference between numbers and one less than the term that they want.
- Knowing how to count is essential for learning algebra and the higher maths, and counting is used for almost everything humans do.


View all lessons
Ordering and comparing numbers beyond 1000
In this lesson, we will look at how to work out whether a number is greater than or less than another number. We will also order and compare numbers.
Activate - 3 Questions
Check - 3 questions
Let's get ready


- Open Access
Generalized Meixner-Pollaczek polynomials
© Kanas and Tatarczak; licensee Springer 2013
- Received: 4 February 2013
- Accepted: 22 April 2013
- Published: 7 May 2013
We consider the generalized Meixner-Pollaczek (GMP) polynomials of a variable and parameters , , , defined via the generating function
We find the three-term recurrence relation, the explicité formula, the hypergeometric representation, the difference equation and the orthogonality relation for GMP polynomials . Moreover, we study the special case of corresponding to the choice and , which leads to some interesting families of polynomials. The limiting case () of the sequences of polynomials is obtained, and the orthogonality relation in the strip is shown.
MSC:33C45, 30C10, 30C45, 39A60.
- Meixner-Pollaczek polynomials
- difference equation
- generating function
- orthogonal polynomials
- Fisher information
The importance of follows from the extremality for the famous Bieberbach conjecture. The Koebe function is univalent and starlike in and maps the unit disk onto the complex plane minus a slit .
was extensively studied by Pommerenke , who investigated a universal invariant family .
The definition of was extended for a nonzero complex number α by Yamashita . The classical result of Hille  ascertains that is univalent in if and only if is in the union A of the closed disks and . Making use of geometric properties, Yamashita  described how tends to be univalent in the whole as α tends to each boundary point of A from outside.
Obviously, we have .
Let denote the moment functional that is a linear map . A sequence of polynomials is an orthogonal polynomials sequence (OPS) with respect to if has degree n, for and for all n.
where , , the numbers and are constants, for and is arbitrary (see [, Ch. I, Theorem 4.1]). The sequences of orthogonal polynomials are symmetric if for all n (see [, Ch. I, Theorem 4.3]) or that in (2.1) are all zero.
Polynomials with exponential generating functions are among the most often studied polynomials. One of them is the Meixner-Pollaczek polynomials. The Meixner-Pollaczek polynomials were first invented by Meixner . The same polynomials were also considered independently by Pollaczek . These polynomials are classified in the Askey-scheme of orthogonal polynomials [6, 11].
Some of the main properties of these polynomials are presented in Erdélyi et al. , Chihara , Askey and Wilson  and in the report by Koekoek and Swarttouw . Detailed analyses with applications of these polynomials are also made by several authors. Among others, the works of Rahman , Atakishiyev and Suslov , Bender et al. , Koornwinder  and the extensive work of Li and Wong  may be included.
This paper is mainly concerned about the generalized Meixner-Pollaczek (GMP) polynomials. We also study the special cases of , corresponding to the choice and , which lead to some interesting families of polynomials.
In this section we find the three-term recurrence relation, the explicité formula, the hypergeometric representation, the difference equation and the orthogonality relation for (GMP) polynomials .
- (a)satisfy the three-term recurrence relation
- (b)are given by the formula(3.1)
- (c)have the hypergeometric representation(3.2)
- (d)Let . The function satisfies the following difference equation:(3.3)
We differentiate the formula (1.2) with respect to z, and after multiplication by , we compare the leading coefficients of .
- (b)The Cauchy product of the power series
- (c)Applying the formula from [, vol.1, p.82],
- (d)Inserting and instead of x into the generating function (1.2), we find that
which together with (3.4) gives (3.3).
Comparing both sides of the above, we get the equality (3.9). □
then (3.10) is a natural consequence. □
Let us consider now the case . We observe that such a case leads to the very interesting family of symmetric polynomials. Some special cases of are known in the literature for . These are the symmetric Meixner-Pollaczek polynomials, denoted by , . For instance, Bender et al.  and Koornwinder  have shown that there is a connection between the symmetric Meixner-Pollaczek polynomials and the Heisenberg algebra. Another example is , where the symmetric Meixner-Pollaczek polynomials are considered.
- (a)the system satisfies
the sequence of polynomials is an orthogonal basis in the Hilbert space ,
the norm of polynomials is if and 1 if .
- (a)By (4.2) we have
- (b)In order to prove the orthogonality of polynomials and compute their norms, it suffices to show that(4.3)
- (c)In the light of (a) and equation (4.3), we have
Comparing the coefficients of the powers of s and , we obtain the desired result. □
around a closed contour K about the origin with radius less than 1.
- (a)The QMP polynomials satisfy the three-term recurrence relation
- (b)The polynomials are given by the formula
- (c)The polynomials have the hypergeometric representation(5.1)
- (d)The polynomials satisfy the following difference equation:
- (e)The polynomials are orthogonal on with the weight
In this work we use the ideas of  to compute the Fisher information of QMP polynomials.
with defined as in (5.3).
Proof For GMP we have .
and the result follows. □
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This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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For Primary Schools
The Mastering Number Programme runs for three terms and prioritises the development of solid number sense, including fluency and flexibility with number facts, which will have a lasting impact on future learning for all children. The programme will involve high quality professional development for teachers.
For schools in the Mastering Number Programme, children in Reception, Year 1 and Year 2 will have a daily teacher-led session of 10 to 15 minutes, designed to ensure that pupils develop fluency with, and understanding of, number that is crucial to future success in maths and academic progress more generally.
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Lead participants from Work Group schools will be three teachers, one each from Reception, Year 1 and Year 2, with some support given to subject leaders and headteachers.
This programme is open to all state-funded primary schools in England. If oversubscribed, priority for places will be given to schools with a high proportion of disadvantaged children.
What is involved?
All lead participants will be supported by central training and resources. There is an expectation that they will provide a daily teaching session for all children of 10 to 15 minutes, in addition to their normal maths lesson. There is also the expectation that they will contribute to an online community to share practice and engage in critical reflection.
In this short video, Debbie Morgan, the NCETM’s Director for Primary, explains how the abacus-like rekenrek can be used to help children develop confidence and fluency with number. Use of a rekenrek is one element of the Mastering Number Programme, and participating schools will be provided with sets of the equipment free of charge.
We’ve produced a flyer summarising the programme – Download it to share with colleagues in your own and other schools.
What will you learn?
- develop skills in working in a professional learning community, reflecting with other colleagues on their own practice, and refining skills through support and challenge within a community
- develop a secure understanding of how to build firm mathematical foundations with a stronger subject and pedagogical understanding for EYFS and KS1
- work to develop intentional teaching strategies focused on developing fluency in calculation and number sense for ALL children
- develop understanding and their use of appropriate manipulatives to support their teaching of mathematical structures
What is the cost?
Mastering Number is fully funded by the Maths Hubs Programme so is free to participating schools.


What is YSPA?
YSPA is a 4-week residential research and enrichment program for 32 rising high school juniors and seniors who are interested in astronomy, physics, math, computer programming, and other science and tech fields. Students take classes in physics, math, and astronomy, complete a scientific research project using telescopes at Yale, learn the basics of computer programming and data science, and present their results at the end of the program. YSPA has a small student-to-faculty ratio (32:6), so students get to know each other and the faculty very well. It’s a summer experience like no other.


Accelerated Projected Gradient Algorithms for Sparsity Constrained Optimization Problems
Jan Harold Alcantara · Ching-pei Lee
Hall J (level 1) #434
Abstract: We consider the projected gradient algorithm for the nonconvex best subset selection problem that minimizes a given empirical loss function under an $\ell_0$-norm constraint. Through decomposing the feasible set of the given sparsity constraint as a finite union of linear subspaces, we present two acceleration schemes with global convergence guarantees, one by same-space extrapolation and the other by subspace identification. The former fully utilizes the problem structure to greatly accelerate the optimization speed with only negligible additional cost. The latter leads to a two-stage meta-algorithm that first uses classical projected gradient iterations to identify the correct subspace containing an optimal solution, and then switches to a highly-efficient smooth optimization method in the identified subspace to attain superlinear convergence. Experiments demonstrate that the proposed accelerated algorithms are magnitudes faster than their non-accelerated counterparts as well as the state of the art.
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