OpenLLM-France/wikimedia huggingface dataset:
# Solutions To Mathematics Textbooks/Principles of Mathematical Analysis (3rd edition) (ISBN 0070856133)/Manual of style

Please use LaTeX to format these pages. Help with displaying formulas and symbols can be found here.
# Solutions To Mathematics Textbooks/Probability and Statistics for Engineering and the Sciences (7th ed) (ISBN-10: 0-495-38217-5)/Chapter 2

Components of a certain type are shipped to a supplier in batches of ten. Suppose that 52% of all such batches contain no defective components, 31% contain one defective component, and 17% contain two defective components. Two components from a batch are randomly selected and tested. What are the probabilities associated with 0, 1, and 2 defective components being in the batch under each of the following conditions? (Round your answers to four decimal places.)
# Chapter 4 - Continuous Random Variables and Probability Distributions
## Section 4.1
### Exercise 1

Given the density function $f(x)={\begin{cases}0.5x&0\leq x\leq 2\\0&{\text{otherwise}}\end{cases}}$
* Part a. Find $P(X\leq 1)$
${\begin{array}{rl}P(X\leq 1)&=\int _{-\infty }^{1}f(x)dx\\&=\int _{0}^{1}0.5xdx=\left.{\frac {x^{2}}{2}}\right|_{0}^{1}\\&={\frac {1}{4}}\end{array}}$
* Part b.
${\begin{array}{rl}P(0.5\leq X\leq 1.5)&=\int _{0.5}^{1.5}f(x)dx=\left.{\frac {x^{2}}{2}}\right|_{0.5}^{1.5}\\&={\frac {1}{2}}\end{array}}$
* Part c.
${\begin{array}{rl}P(1.5<X)&=\int _{0.5}^{2}f(x)dx=\left.{\frac {x^{2}}{2}}\right|_{0.5}^{2}\\&=0.438\end{array}}$

### Exercise 2

Let $X{\text{ be Uniform}}(-5,5)$
* Part a.
${\begin{array}{rl}P(X<0)&=\int _{-5}^{0}{\frac {1}{5--5}}dx\\\\&=\left.{\frac {x}{10}}\right|_{-5}{0}\\\\&={\frac {1}{2}}\end{array}}$
* Part b.
$P(-2.5<X<2.5)=\int _{-2.5}^{2.5}{\frac {1}{10}}dx={\frac {1}{2}}$
* Part c.
$P(-2<X<3)=\int _{-2}^{3}{\frac {1}{10}}dx={\frac {1}{2}}$
* Part d. For $-5<k<k+4<5$, compute
${\begin{array}{rl}P(k<X<k+4)&=\int _{k}^{k+4}{\frac {1}{10}}dx\\\\&=\left.{\frac {x}{10}}\right|_{k}^{k+4}\\\\&={\frac {k+4}{10}}-{\frac {k}{10}}\\\\&={\frac {4}{10}}={\frac {2}{5}}\end{array}}$

### Exercise 3.

Let $f(x)$ be a probability density function.
    $f(x)={\begin{cases}0.09375(4-x^{2})&-2\leq x\leq 2\\0&{\text{otherwise}}\end{cases}}$
* Part a. graph $f(x)$
* Part b.
$P(X>0)=\int _{0}^{2}f(x)dx=0.5$
* Part c.
    $P(-1<X<1)=\int _{-1}^{1}f(x)dx=0.6875$
* Part d.
    $P(X<-0.5{\text{ or }}X>0.5)=1-P(-5\leq X\leq 0.5)=1-\int _{-0.5}^{0.5}f(x)dx=1-0.3671888=0.632812$

### Exercise 4.

Let $X$ have the Rayleigh distribution with the probability density function
$f(x)={\begin{cases}{\frac {x}{\theta ^{2}}}e^{-x^{2}/\left(2\theta ^{2}\right)}&x>0\\&\\0&{\text{otherwise}}\end{cases}}$
* Part a.
Verify that $f(x)$ is a pdf.
** First notice that $f(x)\geq 0$ for all $x$
** Next show the integral over the whole number line equals one:
       ${\begin{array}{rll}\int _{-\infty }^{\infty }f(x)dx&=\int _{0}^{\infty }{\frac {x}{\theta ^{2}}}e^{-x^{2}/\left(2\theta ^{2}\right)}dx&{\text{let }}y=x/\theta ,dy=1/\theta dx\\\\&=\int _{0}^{\infty }ye^{-y^{2}/2}dy&\\\\&=\left.e^{-y^{2}/2)\right|_{0}^{\infty }\\\\&=0--1\\&=1\end{array}}$
* Part b. Let $\theta =100$.
** Probability $X$ is at most 200
       $P(X\leq 200)=\int _{0}^{200}f(x)dx=1-{\frac {1}{e^{2}}}\approx 0.864665$
** Probability $X$ is less than 200
       $P(X<200)=\int _{0}^{200}f(x)dx=1-{\frac {1}{e^{2}}}\approx 0.864665$
** Probability $X$ is at least 200
       $P(X\geq 200)=1-P(X<200)={\frac {1}{e^{2}}}\approx 0.135335$
* The probability $X$ is between 100 and 200 assuming $\theta =100$.
    $P(100<X<200)=\int _{100}^{200}f(x)dx={\frac {e^{3/2}-1}{e^{2}}}\approx 0.471195$
* Give an expression for $P(X\leq x)$, i.e., define the cumulative distribution function.
    $P(X\leq x)=\int _{-\infty }^{x}f(y)dy={\begin{cases}0&x\leq 0\\1-\exp \left({\frac {-x^{2}}{2\theta ^{2}}}\right)&x>0\end{cases}}$
# Solutions To Mathematics Textbooks/Probability and Statistics for Engineering and the Sciences (7th ed) (ISBN-10: 0-495-38217-5)

== Solutions for Probability and Statistics for Engineering and the Sciences, 7th ed, by Jay L. Devore

## Contents

Chapter 1 - Overview and Descriptive Statistics
Chapter 2 - Probability
Chapter 3 - Discrete Random Variables and Probability Distributions
Chapter 4 - Continuous Random Variables and Probability Distributions
Chapter 5 - Joint Probability Distributions and Random Samples
Chapter 6 - Point Estimation
Chapter 7 - Statistical Intervals Based on a Single Sample
Chapter 8 - Tests of Hypotheses Based on a Single Sample
Chapter 9 - Inferences Based on Two Samples
Chapter 10 - The Analysis of Variance
Chapter 11 - Multifactor Analysis of Variance
Chapter 12 - Simple Linear Regression and Correlation
Chapter 13 - Nonlinear and Multiple Regression
Chapter 14 - Goodness-of-Fit Tests and Categorical Data Analysis
Chapter 15 - Distribution-Free Procedures
Chapter 16 - Quality Control Methods
# Contents

Chapter 1 - Informal Logic
Chapter 2 - Strategies for Proofs
Chapter 3 - Sets
Chapter 4 - Functions
Chapter 5 - Relations
Chapter 6 - Infinite and Finite Sets
Chapter 7 - Selected Topics
Chapter 8 - Number Systems
Chapter 9 - Explorations
# Super Mario World/Special/Funky

Funky is the last level in Special. Once you clear it, the entire game will look very different. Koopa will appear as a human figure and Bullet Bill will appear as a black bird in every level. You can save your progess after clearing this level. While this level isn't that hard, you only get 200 seconds to finish it! It really is a long course! Bring Yoshi along for the ride and get a cape, you'll need it! With Yoshi, eat the green berrys to add more time to this simple yet annoying course. Once you beat there's a message from the games designers! A cool trick: Get close to the first green fruit with Yoshi. When the timer hits 99 eat the fruit, then rush to the next fruit and wait on the timer again. The music will go faster and faster and faster and faster and faster and you get the point. Play that funky music!
# Super Mario World/Special/Outrageous

Its name also describes the difficulty. It is arguably the most difficult level in the game! Have good timing skills, bring Yoshi, have a cape! This is the first (and last?) time you're going to meet walking fire in this game! Have fun!
In order to pass this level you have to feed Yoshi the second spring that's among the bullet bill cannons. To pass the cannon that is right next to the spring board you have to jump on a bullet bill at the right time. Be hasty while clearing the rest of the cannons, otherwise Yoshi will swallow the spring.
Probably the easiest (and least satisfying) way of passing this level is to fly over all the action - just make sure to clear a column of blocks about halfway through
# Super Mario World/Special/Way Cool

Way Cool is the third level in Special. It consists of many mechanical lifts and switches. You can use the Yoshi Wings with Groovy's Yoshi Block for a quick blue yoshi for the Funky Music Code.
# Super Mario World/Small status

When Mario is small, he is called small Mario.

## Getting Small Mario

* When starting a new game or when restarting a saved one, the player begins with small Mario.
* If cape Mario, fire Mario or super Mario get hit they turn into small mario.
* If Mario dies, he restarts as small mario.

## Abilities

Small mario is the weakest Mario.
* A single hit kills small Mario.
* Small mario cannot break yellow bricks using spin jump.
* However, his small size allows him to access areas that are otherwise more difficult to access.

## Misc

It is quite possible, but more difficult, to beat the game using only small mario. In fact 90 of the 96 exits can be beaten using only small mario. The exceptions are:
* Secret exit of Donut Ghost House, cape Mario required
* Secret exist of Cheese Bridge Area, Yoshi or cape Mario required
* Secret exit of Forrest of Illusion 3, super Mario required
* Secret exit of Valley of Bowser 4, Yoshi required
* Both exits of Star Road 1, super Mario required
# Super Mario World/Special/Mondo

Bring Yoshi and a fire power up with the cape power up as a backup, you'll need all the help you can get with this level, not the hardest course in the game, but one of the most annoying.
# Super Mario World/Special/Awesome

This course moves away from the norm we are used to seeing in this game. The koppa troppas have command over this level and can easily kill you if you don't have split second timing. This is the only part of the game where you're gonna see ice blocks, light blue pipes and fancy little platforms. Watch out for the flying fishes we seen in Super Mario Brothers back in the 80s
Several unshelled Koopa Troopas await with a shell atop U-shaped chasms. If you run to the right side of the chasm and wait, the shell the Koopa throws won't return. When you arrive at a pit with P-Switch on a corner, don't activate the switch. Carry it with you until you see a coin plataform and a [?] block. The block contains a star, which will help a lot during the next stage.
# Super Mario World/Special/Gnarly

Gnarly is the first level in the Special World.

## Description

The level is generally a level in which you must climb vines and jump well to continue. It is one of the easier levels of the Special World.
This is a very fun level. You can even beat it if you have Yoshi. Start with Yoshi, go all the way to the right and eat the blue-shelled Koopa on the left. Then fly up all the way to the left to earn one (1) extra life.
When you get to the top of the level, fall to the far left side, a 1-up mushroom will appear from the musical note blocks.
# Super Mario World/Special/Groovy

You've seen this course before, it is basically the Super Mario World demo video played just before you head into the file menu screen. This course is quite simple to play. Also good for Blue Yoshis when combined with Way Cool's Yoshi Wings. There is a changing block here.
# Super Mario World/Special/Tubular

Tubular :
| Level type | Elevated |
| Entrance | Gnarly |
| Exit | Way Cool |
| Secret Exit | None |
| Time at start | 300 |
| Difficulty | Ridiculous |

## Info

This is the most frustrating level in the game and impossible if you don't know what to do. The main means of completion is through the use of balloons. However, it is possible to complete the level by using a blue Yoshi and swallowing a Koopa shell, but a sufficient amount of blue Koopa shells do not exist to beat the level with a regular Yoshi.

## Walkthrough

Jump onto and past the Chuck ahead. Go past the next pipe, but watch out for the piranha, and run and jump into the Dragon Coin and over the Chuck. Go over the yellow pipes and activate the P-Switch. There are two ways to collect the P-Balloon. One: quickly drop down and left onto the springboard, and hit the [?] block and collect the P-Balloon that comes from it. Two: Jump to the right side of the [?] block and do a spin jump while on you're in the air, the P-Balloon will come out of that side and you will collect it inmeadiately, which will allow you to avoid enemies. Float and get the Dragon Coin that's under the block, and go up and collect the Dragon Coin above the Chuck on the block. Hit the ? block just to the right and renew your inflatingness with the P-Balloon. Head through the flying Koopas and collect the Dragon Coin that are between the volcano flowers when the coast is clear. Go onward, past the yellow pipe and under the Chuck so you can get another P-Balloon. Now watch out for footballs from the above Chuck when getting it, because unless you are prepared, one WILL hit you. I recommend that you wait until one has been throw and gone, then hit the block and get the P-Balloon. Hit the block that the Chuck above is resting on (for safety sake, and for revenge) and go under the next to volcano flowers, and onto the exit. Save.
# Super Mario World/Special

If you found the secret exit in Star World 5, you can access the secret world Special. Be careful though, the levels in Special are far from easy.
Once Special is completed, the colors of the world map, as well as some enemies, will be different. The upper Star Road will lead you back to Yoshi's House.
Special's levels:
* Gnarly
* Tubular
* Way Cool
* Awesome
* Groovy
* Mondo
* Outrageous
* Funky
Note: When completed, this page will contain info on the creatures in and the difficuty of the world.
# The Computer Revolution/Communication/Information age

The information age became more aware as the development of computers progressed. People initially questioned the idea of computers but soon realized that computers would pave the future of society and the economy. As people became aware of the practical use of computers a few individuals took much needed risks and began to explore the confines generalized of what a computer could and could not do. As computers were exploited new horizons emerged; innovations began to flourish. Ideas of mass storage and world wide communication were soon turned into reality .
As time progressed forward and knowledge of computers was solidified. Simple basic operations were surpassed with an every lying quest for more. The World Wide Web was created and the basic momentum of computers exploded exponentially as millions of people shared idea’s generating new foundations and relationships for every aspect of society as a whole. Not only had the computer become a revolution, it had laid out the very basic blue prints for a revolution of an entire world.
# The Computer Revolution/Chapter 1

How Computers Work: 3 Key components
1. The purpose of the Computer: - Basically to turn raw data into useful information
Data: consists of the raw facts and figures that are processed into information, i.e., the votes for a presidential election, which are used to decide who has won.
Information: is data that has been summarized or otherwise manipulated for use in decision making.
2. The difference between Hardware and Software.....
Hardware: consists of all the machinery and equipment in a computer system, i.e. the mouse, monitor, hard drive, keyboard etc
Software: "programs" all the electronic instructions that tell the computer how to perform a task. Such as a CD ROM, with instructions and or setup programs.
3. The basic Operations of a Computer:
* Input operation: Is whatever is put into the computer system
* Processing operation: The manipulation a Data into information
* Storage operation: "memory"
** Primary Storage- holds data waiting to be processed
** Secondary Storage- Refers to devices that store data/info permanently
4. Output Operation: Basically what the results of processing, usually info
5. Communications operation: Online Communications, email, MSN, etc.
# The Computer Revolution/Chapter 2

Internet Connections

## Telephone

    Pluses- Inexpensive available everywhere
    Minuses- Slow, connection only supports only a single user

## ISDN

    Pluses- Faster than Dial-Up,, uses conventional phone lines
    Minuses- More expensive than dial up, no longer supported by phone companies for single users, mostly uses by small businesses

## DSL

    Pluses- Can support many users, uploads faster than cable
    Minuses- Needs to be close to phone switching station, limited choice of service providers, supports only a single user

## T1 line

    Pluses- reliable high speed downloading and uploading, you can talk and transmit at the same time
    Minuses- Expensive best for businesses

## Cable Modem

    Pluses- Fast, always on, broadband connections, downloads faster than DSL
    Minuses- Slower service during peak periods, vulnerable to outside intrusion, limited choices of providers, not always available to businesses

## Satellite

    Pluses- Wireless, fast, reliable, always on, goes where DSL and Cable cant
    Minuses-High setup and monthly costs, must have unobstructed view of southern sky

## Wi-Fi

    Pluses- uses a beefed up version of the current cellphone network, ultimately should be available everywhere
    Minuses- Range of access is usually only 50-300 feet (to access point)

## 3G

    Pluses- Functions like Wi-Fi but without the need for hot spots, uses existing cellphone network
    Minuses- Low Battery power on some models, phones are relatively large
# The Computer Revolution/Communication/Exterior

Customizing has progressed from the simple replacement of internal components to creating a custom design or alternative case for holding essentials parts. Recently this trend has developed as users try to create a personal environment that exceeds the interior of the computer and surrounds them. Initially starting with the addition of colors produced by the manufactures, people started to add stickers to the PC's exterior tower and along the screen. As more people became aware of the potential of turning a factory PC into a PC which represents them .
The degree of customizing moved farther away from conventional means and more into a science. Stickers became all too common and soon the exterior of the tower was being cut and molded different shapes, these shapes take up different forms such as emblems of certain games or a more machine orientated look. Soon LED lights became more common illuminating glass panels or water driven cooling systems. Followed by the addition composite materials completely replacing the exterior shell resembling alien environments with unique paint schemes.
Companies soon became aware of the creativity that people were investing into their personal computers. Companies such as Alienware soon started to develop unique casings to appeal to the demand of a unique computer that as more than just hardware but also visual as well. As customizing becomes more personal for each person , computers become less uniform and more individual reflecting the persons perception of the ideal computer.
People have become very creative when it comes to the customization of their desktop computers. For example they can install cool light features into the monitor of their computers to provide a cool looking computer.lighting modifications to a customized desktop computer are much more enjoyable when they can be seen, and as a result, case windows have become a common modification. Also, large window panes can be cut into the side of the customized desktop computer, then lined with acrylic glass to allow the lighting effects to shine out. Then there are some computers where the creativity is blown through the roof ad very well done. For example some people install beer taps in them. When it comes to computer customization the sky is the limit.
# The Computer Revolution/Communication/How communications help with education

Computers have become an essential necessity concerning education in the today’s world. Examples of this would be the use of: power point presentations in classrooms, Blackboard, email, and editing.
May professors use power point presentations to lecture a particular class, its makes it much easier for the professor to teach a subject with power point slides.
Also, the use of the online Blackboard has enabled professors to communicate much more effectively. Students these days are enabled to complete assignments and hand them in online via Blackboard. This makes it more convenient to complete assignments from anywhere; from home, work, or even across the world. The use of email also does the same thing.
Editing assignments has become easier for students because of computers. The use of spell and grammar checkers like in Microsoft Word make editing a lot faster than before. Also, this saves paper and time. The students no longer have to print out rough copies and re-write assignment that require corrections; rather they just edit everything and print out the final copy once all the editing is completed.
# Usability for Nerds/Involve users in the design

Potential users should be involved from the very beginning of the design process. There are two kinds of users that you can involve:
Experienced users. This is the kind of users that have long time experience with using a similar product (a previous version of your product or a competing product). Experienced users often have many suggestions for things that can be improved.
Novice users. This is the kind of persons that might need your product but have never used a similar product. They are useful for asking the silly questions that you haven't thought about. How do I ... ?
Both types of users can be involved in usability tests as soon as you have the first prototype or even just a drawing of the user interface.
← User profile · Usability test →
# Usability for Nerds/Hardware/Mouse

A mouse is a wonderful device to make an interface easy to use for beginners because it enables the user to select or manipulate visual objects. But a mouse has severe disadvantages when it comes to ergonomics. Many people get stress and strain injuries when they have to make precise movements with the mouse. The problems get worse if a ball-mouse gets dirty or an optical mouse is used on a too homogeneous surface so that it reacts less reliably.
However, some pointer devices may use accelerometers to track their movements. They are much more resistant against external conditions, but usually offer less precision.
Double-clicking is particularly a problem because many systems don't accept even the slightest movement of the mouse between the two clicks. Making a fast movement in one direction (mouse button down and up twice) and avoiding even the slightest movement in another direction (horizontally) is a stressing task that puts extraordinary demands on your dexterity, and some people are simply unable to do it. All systems should be set up to accept a certain movement between the two clicks or, preferably, have an extra mouse button instead of the double-click.
A lot of alternative pointing devices have been suggested: joysticks, trackballs, pens, touch pads, touch screens, etc., but no perfect solution has so far been found.
The mouse cursor must respond to mouse movements with negligible delay. If there is a delay, and especially if the delay is unpredictable, then the user does not know where they are clicking. This can happen if the underlying code has lazy loading or lazy binding. It can also happen when a computer is controlled remotely over a network. It is usually impossible to obtain a sufficiently low and consistent delay when remote controlling a computer over a network. Therefore, systems that can be remote controlled must either have enough functionality at the client side to generate the mouse feedback, or it must use an interaction scheme with no mouse.
All systems should have shortcut keys for all common commands in order to enable users to use the keyboard rather than the mouse as much as possible. Using the keyboard is generally faster than using the mouse if the user can remember the commands. Ideally, all software should be operable without a mouse, with the possible exception of manipulating graphics.
← Keyboard · Screen →
# Usability for Nerds/Hardware/Sound

Humans are used to hearing sounds from everything that we touch and move; and it is natural to use the sound as feedback whenever we use mechanical tools and machines.
Computers and software can generate sounds in order to imitate the sound effects of physical objects. However, these sound effects are often more disturbing than helpful. Have you ever been in a classroom where all the students have their own computer, and all the computers play a jingle when turned on? Then you know what I mean.
Sound effects should be used as little as possible because they can be annoying, especially to other people nearby. Most sound effects should be off by default. There are cases, however, where sound effects are useful, especially when it is necessary to catch the user's attention to some problem or as a confirmation of whether some event has actually occurred, much like tactile feedback from the keyboard.
Sound effects are of course useful in video games. The user may use earphones in order to avoid disturbing other people nearby.
← LEDs and displays · Alarms →
# Usability for Nerds/Hardware

* Keyboard
* Mouse
* Screen
* LEDs and displays
* Sound
* Alarms