# DISCRETE RANDOM VARIABLES

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 DISCRETE RANDOM VARIABLES DISCRETE RANDOM VARIABLES Documents prepared for use in course B , New York University, Stern School of Business Definitions page 3 Discrete random variables are introduced here. The related concepts of mean, epected value, variance, and standard deviation are also discussed. Binomial random variable eamples page 5 Here are a number of interesting problems related to the binomial distribution. Hypergeometric random variable page 9 Poisson random variable page 15 Covariance for discrete random variables page 19 This concept is used for general random variables, but here the arithmetic for the discrete case is illustrated. edit date 8 JAN 007 Gary Simon, 007 Cover photo: Trout hatchery, Connetquot Park, Long Island, March 006

2 DISCRETE RANDOM VARIABLES

3 Discrete Random Variables and Related Properties {{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{ Discrete random variables are obtained by counting and have values for which there are no in-between values. These values are typically the integers 0, 1,,. Random variables are usually denoted by upper case (capital) letters. The possible values are denoted by the corresponding lower case letters, so that we talk about events of the form [X = ]. The random variables are described by their probabilities. For eample, consider random variable X with probabilities P[X = ] You can observe that the probabilities sum to 1. The notation P() is often used for P[X = ]. The notation f() is also used. In this eample, P(4) = The symbol P (or f) denotes the probability function, also called the probability mass function. The cumulative probabilities are given as F ( ) = Pi ( ). The interpretation is that F() is the probability that X will take a value less than or equal to. The function F is called the cumulative distribution function (CDF). This is the only notation that is commonly used. For our eample, F(3) = P[X 3] = P[X=0] + P[X=1] + P[X=] + P[X=3] i = = 0.75 One can of course list all the values of the CDF easily by taking cumulative sums: P[X = ] F() The values of F increase. The epected value of X is denote either as E(X) or as μ. It s defined as. The calculation for this eample is E( X) = P( ) = P[ X = ] E(X) = = =.80 This is also said to be the mean of the probability distribution of X. { page 3 gs003

4 Discrete Random Variables and Related Properties {{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{ The probability distribution of X also has a standard deviation, but one usually first defines the variance. The variance of X, denoted as Var(X) or σ, or perhaps σ, is Var(X) = ( μ) P( ) = ( μ ) P( X = ) This is the epected square of the difference between X and its epected value, μ. We can calculate this for our eample: ( -.8) P[X = ] (-.8) P[X = ] The variance is the sum of the final row. This value is This is not the way that one calculates the variance, but it does illustrate the meaning of the formula. There s a simplified method, based on the result ( μ ) P[ X = ] = P[ X = ] μ found μ, and the sum P[ X = ] is fairly easy to calculate. For our eample, this. This is easier because we ve already sum is = Then ( μ) P [ X = ] = = = This is the same number as before, although obtained with rather less effort. The standard deviation of X is determined from the variance. Specifically, SD(X) = σ = Var ( X ). In this situation, we find simply σ = It should be noted that random variables also obey, at least approimately, a variant on the empirical rule used with data. Specifically, for a random variable X with mean μ and standard deviation σ, we have P[ μ - σ X μ + σ ] 3 P[ μ - σ X μ + σ ] 95% X { page 4 gs003

5 BINOMIAL RANDOM VARIABLES 1. Suppose that you are rolling a die eight times. Find the probability that the face with two spots comes up eactly twice. SOLUTION: Let X be the number of successes, meaning the number of times that the face with two spots comes up. This is a binomial situation with n = 8 and p = 1 6. The probability of eactly two successes is P[ X = ] = ( 6) ( 6) = 8. This can 8 6 be done with a calculator. There are various strategies to organize the arithmetic, but the answer certainly comes out as about y Some calculators have keys like, and these can be useful to calculate epressions of the form 6 8. Of course, 6 8 can always be calculated by careful repeated multiplication. The calculator in Microsoft Windows will find 6 8 through the keystrokes 6, y, 8, =.. The probability of winning at a certain game is If you play the game 10 times, what is the probability that you win at most once? SOLUTION: Let X be the number of winners. This is a binomial situation with n = 10 and p = We interpret win at most once as meaning X 1. Then P[ X 1 ] = P[ X = 0 ] + P[ X = 1 ] = = = = ( ) = If X is binomial with parameters n and p, find an epression for P[ X 1 ]. SOLUTION: This is the same as the previous problem, but it s in a generic form. n n n 1 n P[ X 1 ] = P[ X = 0 ] + P[ X = 1 ] = p ( 1 p) + p ( 1 p) 1 n n = ( 1 p) + np ( 1 p) 1 n 1 = ( 1 p) (( 1 p) + np) ( ) n 1 = ( 1 ) 1 + ( 1) p n p { page 5 gs003

7 BINOMIAL RANDOM VARIABLES searching for the smallest k for which P[ X k-1] Here is a set of cumulative probabilities calculated by Minitab: P[ X ] The first time that the cumulative probabilities cross 0.99 occurs for = 14. This corresponds to k-1, so we report that k = 15. The smallest number of defects which would cause a 1% rare-event alert is If you flip a fair coin 19 times, what is the probability that you will end up with an even number of heads? SOLUTION: Let X be binomial with n = 19 and p = 1. This seems to be asking for P[ X = 0 ] + P[ X = ] + P[ X = 4 ] +. + P[ X = 18 ], which is an annoying calculation. However, we ve got a trick. Consider the first 18 flips. The cumulative number of heads will either be even or odd. If it s even, then the 19 th flip will preserve the even total with probability 1. If it s odd, then the 19 th flip will convert it to even with probability 1. At the end, our probability of having an even number of heads must be 1. This trick only works when p = Suppose that you are playing roulette and betting on a single number. Your probability of winning on a single turn is You would like to get at least three winners. Find the minimum number of turns for which the probability of three or more winners is at least SOLUTION: The problem asks for the smallest n for which P[ X 3 ] Since Minitab computes cumulative probabilities in the form, we ll convert this question to finding the smallest n for which P[ X ] 0.0. This now requires a trial-and-error search. It helps to set up a diagram in which the first row is for an n that s likely to be too small and a last row for an n that is likely to be too big. n P[ X ] value of n is too small too large { page 7 gs003

8 BINOMIAL RANDOM VARIABLES Then intervening positions can be filled in. Let s try n = 100. This would result in P[ X ] = , revealing that n = 100 is too small. The table gets modified to this: n P[ X ] value of n is too small too small too large Now try n = 150, getting P[ X ] = This says that n = 150 is too small, but not by much. Here s what the table looks like: n P[ X ] value of n is too small too small too small After a little more effort, we get to this spot: too large n P[ X ] value of n is too small too small too small too small too small just right! too large For n = 16, the cumulative probability drops below 0.0 for the first time. This is the requested number of times to play the game. { page 8 gs003

9 THE HYPERGEOMETRIC RANDOM VARIABLE n Recall that the binomial coefficient r is used to count the number of possible selections of r things out of n. Using n = 6 and r = would provide the number of possible committees that could be obtained by selecting two people out of si. n n! The computational formula is r =. For n = 6 and r =, this would give r! ( n r)! 6 6! = = = 6 5 = 15. If the si people are named! 4! ( 1 ) ( ) 1 A, B, C, D, E, and F, these would be the 15 possible committees: A B B C C E A C B D C F A D B E D E A E B F D F A F C D E F There are a few useful manipulations: 0! = 1 (by agreement) n r = n n r 6 6 In the committee eample, = 4, so that the number of selections of two people to be on the committee is eactly equal to the number of selections of four people to leave off the committee. n 0 = n n = 1 n 1 = n n = ( 1) n n { page 9 gs003

10 THE HYPERGEOMETRIC RANDOM VARIABLE The hypergeometric distribution applies to the situation in which a random selection is to be made from a finite set. This is most easily illustrated with drawings from a deck of cards. Suppose that you select five cards at random from a standard deck of 5 cards. This description certainly suggests a card game in which five cards are dealt to you from a standard deck. The game of poker generally begins this way. The fact that cards will also be dealt to other players does not influence the probability calculations, as long as the identities of those cards are not known to you. You would like to know the probability that your five cards (your hand ) will include eactly two aces. The computational logic proceeds along these steps: 5 * There are 5 possible selections of five cards out of 5. These selections are equally likely. 4 * There are four aces in the deck, and there are ways in which you can identify two out of the four. 48 * There are 48 non-aces in the deck, and there are 3 ways in which you can identify three of these non-aces. * The number of possible ways that your hand can have eactly two aces 4 48 and eactly three non-aces is 3. This happens because every selection of the aces can be matched with every selection of the non-aces * The probability that your hand will have eactly two aces is. 5 5 The computation is not trivial. If you are deeply concerned with playing poker, the 5 5! number 5 will come up often. It s = =,598,960. 5! 47! Now note that = 6, 3 = = 17,96. Then you can find the desired 3 1 probability 6 17, This is about 4%.,598,960 { page 10 gs003

11 THE HYPERGEOMETRIC RANDOM VARIABLE This technology can be generalized in a useful random variable notation. We will let random X be the number of special items in a sample of n taken at random from a set of N. You will sometimes see a distinction between sampling with replacement and sampling without replacement. The issue comes down to whether or not each sampled object is returned to the set of N before the net selection is made. (Returning an item to the set of N would make it possible for that item to appear in the sample more than once.) In virtually all applications, the sampling is without replacement. The sampling is usually done sequentially, but it does not have to be. In a card game, the same probabilities would apply even if you were given all your cards in a single clump from the top of a well-shuffled deck. Of course, dealing out cards in clumps violates the etiquette of the game. The process is sometimes described as taking a sample of n from a finite population of N. We then use these symbols: N population size n sample size D number of special items in the population N D number of non-special items in the population X (random) number of special items in the sample This table lays out the notation: General notation Card eample Population size N 5 (cards in deck) Sample size n 5 (cards you will be dealt) Special items in the population D 4 (aces in the deck) Non-special items in the population N D 48 (non-aces in the deck) Random number of special items in the sample X Number of special items in your hand (we asked for this to be in the eample) The probability structure of X is given by the hypergeometric formula P[ X = ] = D N D n N n { page 11 gs003

12 THE HYPERGEOMETRIC RANDOM VARIABLE For our eample, this was In a well-formed hypergeometric calculation the numerator upper numbers add to the denominator upper number (as = 5) the numerator lower numbers add to the denominator lower number (as + 3 = 5) As an interesting curiousity, it happens that we can write the hypergeometric probability in the alternate form n N n D P[ X = ] = N D In our eample, this would be 5 5 = The program Minitab, since release 13, can compute hypergeometric probabilities. Suppose that you would like to see the probabilities associated with the number of spades that you get in a hand of 13 cards. The game of bridge starts out by dealing 13 cards to each player, so this question is sometimes of interest to bridge players. In column 1 of Minitab, lay out the integers 0 through 13. You can enter these manually, or you can use this little trick: Calc Make Patterned Data Simple Set of Numbers On the resulting panel, enter the information indicated... Store patterned data in: (enter C1) From first value: (enter 0) To last value: (enter 13) { page 1 gs003

13 THE HYPERGEOMETRIC RANDOM VARIABLE Minitab can then easily find all the probabilities at once: Calc Probability Distributions Hypergeometric Fill out the resulting panel to look like this: The first 13, successes in population, corresponds to our symbol D. (It s unfortunate that Minitab used the letter X for this purpose.) The second 13, sample size, corresponds to our symbol n (and Minitab agrees). When you click OK, the entire column of probabilities appears in column C. Here are some typical problems. Eample 1: What is the most likely number of spades that you will get in a hand of 13 cards? Solution: If you eamine the output that Minitab produced, you ll see P[ X = ] = P[ X = 3 ] = P[ X = 4 ] = All the other probabilities are much smaller. Thus, you re most likely to get three spades. { page 13 gs003

14 THE HYPERGEOMETRIC RANDOM VARIABLE Eample : Suppose that a shipment of 100 fruit crates has 11 crates in which the fruit shows signs of spoilage. A quality control inspection selects 8 crates at random, opens these selected crates, and then counts the number (out of 8) in which the fruit shows signs of spoilage. What is the probability that eactly two crates in the sample show signs of spoilage? Solution: Let X be the number of bad crates in the sample. This is a hypergeometric random variable with N = 100, D = 11, n = 8, and we ask P[ X = ]. This probability is The arithmetic is possible, but it s annoying. Let s use Minitab for this. The detail panel should be this: Minitab will produce this information in its session window: Probability Density Function Hypergeometric with N = 100, M = 11, and n = 8 P( X = ) The requested probability is { page 14 gs003

15 THE POISSON RANDOM VARIABLE The Poisson random variable is obtained by counting outcomes. The situation is not governed by a pre-set sample size, but rather we observe over a specified length of time or a specified spatial area. There is no conceptual upper limit to the number of counts that we might get. The Poisson would be used for The number of industrial accidents in a month The number of earthquakes to strike Turkey in a year The number of maple seedlings to sprout in a 10 m 10 m patch of meadow The number of phone calls arriving at your help desk in a two-hour period The Poisson has some similarities to the binomial and hypergeometric, so we ll lay out the essential differences in this table: Binomial Hypergeometric Poisson no concept of Number of trials n n sample size Infinite (trials could no concept of Population size N go on indefinitely) population size D no concept of event Event probability p N probability Event rate λ no concept of event no concept of event rate rate The Poisson probability law is governed by a rate parameter λ. For eample, if we are dealing with the number of industrial accidents in a month, λ will represent the epected rate. If X is this random variable, then the probability law is P[ X = ] = e λ λ! This calculations can be done for = 0, 1,, 3, 4, There is no upper limit. If the rate is 3. accidents/month, then the probability that there will be eactly two accidents in any month is P[ X = ] = e ! Minitab can organize these calculations easily. In a column of the data sheet, say C1, enter the integers 0, 1,, 3, 4,., 10. It s easy to enter these directly, but you could also use Calc Make Patterned Data. Then do Calc Probability Distributions Poisson. { page 15 gs003

16 THE POISSON RANDOM VARIABLE The information panel should then be filled as indicated: The probabilities that result from this operation are these: Accidents Probability { page 16 gs003

17 THE POISSON RANDOM VARIABLE Here is a graph of the probability function for this Poisson random variable: 0. Probability Accidents This is drawn all the way out to 0 accidents, but it s clear that nearly all the probability action is below 1. EXAMPLE: The number of calls arriving at the Swampside Police Station follows a Poisson distribution with rate 4.6/hour. What is the probability that eactly si calls will come between 8:00 p.m. and 9:00 p.m.? SOLUTION: Let X be the random number arriving in this one-hour time period. We ll use λ = 4.6 and then find P[ X = 6 ] = e ! EXAMPLE: In the situation above, find the probability that eactly 7 calls will come between 9:00 p.m. and 10:30 p.m. SOLUTION: Let Y be the random number arriving during this 90-minute period. The Poisson rate parameter epands and contracts appropriately, so the relevant value of λ is = 6.9. We find P[ Y = 7 ] = e ! The Poisson random variable has an epected value that is eactly λ. The standard deviation is λ. { page 17 gs003

18 THE POISSON RANDOM VARIABLE EXAMPLE: If X is a Poisson random variable with λ = 5, would it be unusual to get a value of X which is less than 190? SOLUTION: If we were asked for the eact number, we d use Minitab to find P[ X 189 ] This suggests that indeed it would be unusual to get an X value below 190. However, we can get a quick approimate answer by noting that E X = mean of X = λ = 5, and SD(X) = λ = 5 = 15. The value 190 is standard deviations below the mean; yes, it would be unusual to get a value that small. This chart summarizes some relevant facts for the three useful discrete random variables. Random variable Binomial Hypergeometric Poisson Description Number of successes in n independent trials, each having success probability p Number of special items obtained in a sample of n from a population of N containing D special items Number of events observed over a specified period of time (or space) at event rate λ Probability function P[ X = ] n p p ( 1 ) n D N D n N n e λ λ! Epected value (Mean) Standard deviation np np ( 1 p) D D D N n n 1 n N N N 1 N λ λ { page 18 gs003

19 COVARIANCE FOR DISCRETE RANDOM VARIABLES Suppose that the two random variables X and Y have this probability structure: Y = 1 Y = X = X = X = We can check that the probability sums to 1. The easiest way to do this comes in appending one row and one column to hold totals: Y = 1 Y = Total X = X = X = Total Thus P(Y = 1) = 0.34 and P(Y = ) = Then E Y = = 1.66 = μ Y E Y = =.98 σ Y = Var(Y) = E Y - ( E Y ) = = = 0.44 σ Y = SD(Y) = Using P(X = 8) = 0.30, P(X = 10) = 0.60, and P(X = 1) = Then E X = = 9.6 = μ X E X = = 93.6 σ X = Var(X) = E X - ( E X ) = = = 1.44 σ X = SD(X) = 1.44 = 1. { page 19 gs003

20 COVARIANCE FOR DISCRETE RANDOM VARIABLES Let s introduce the calculation of Covariance(X, Y) = Cov(X, Y). This is defined as Cov(X, Y) = E[ (X μ X )(Y μ Y ) ] but it is more easily calculated as Cov(X, Y) = E[ X Y ] μ X μ Y Here E[ X Y ] = = Then Cov(X, Y) = = = We can then find the correlation of X and Y as ρ = Corr(X, Y) = Cov X, Y σ σ X a Y f = { page 0 gs003

### Binomial random variables (Review)

Poisson / Empirical Rule Approximations / Hypergeometric Solutions STAT-UB.3 Statistics for Business Control and Regression Models Binomial random variables (Review. Suppose that you are rolling a die

### Random variables, probability distributions, binomial random variable

Week 4 lecture notes. WEEK 4 page 1 Random variables, probability distributions, binomial random variable Eample 1 : Consider the eperiment of flipping a fair coin three times. The number of tails that

### Binomial random variables

Binomial and Poisson Random Variables Solutions STAT-UB.0103 Statistics for Business Control and Regression Models Binomial random variables 1. A certain coin has a 5% of landing heads, and a 75% chance

### Chapter 4 Lecture Notes

Chapter 4 Lecture Notes Random Variables October 27, 2015 1 Section 4.1 Random Variables A random variable is typically a real-valued function defined on the sample space of some experiment. For instance,

### MATH 201. Final ANSWERS August 12, 2016

MATH 01 Final ANSWERS August 1, 016 Part A 1. 17 points) A bag contains three different types of dice: four 6-sided dice, five 8-sided dice, and six 0-sided dice. A die is drawn from the bag and then rolled.

### Expectation Discrete RV - weighted average Continuous RV - use integral to take the weighted average

PHP 2510 Expectation, variance, covariance, correlation Expectation Discrete RV - weighted average Continuous RV - use integral to take the weighted average Variance Variance is the average of (X µ) 2

### The Binomial Probability Distribution

The Binomial Probability Distribution MATH 130, Elements of Statistics I J. Robert Buchanan Department of Mathematics Fall 2015 Objectives After this lesson we will be able to: determine whether a probability

### Chapter 3: DISCRETE RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS

Chapter 3: DISCRETE RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS Part 4: Geometric Distribution Negative Binomial Distribution Hypergeometric Distribution Sections 3-7, 3-8 The remaining discrete random

### Random Variable: A function that assigns numerical values to all the outcomes in the sample space.

STAT 509 Section 3.2: Discrete Random Variables Random Variable: A function that assigns numerical values to all the outcomes in the sample space. Notation: Capital letters (like Y) denote a random variable.

### For a partition B 1,..., B n, where B i B j = for i. A = (A B 1 ) (A B 2 ),..., (A B n ) and thus. P (A) = P (A B i ) = P (A B i )P (B i )

Probability Review 15.075 Cynthia Rudin A probability space, defined by Kolmogorov (1903-1987) consists of: A set of outcomes S, e.g., for the roll of a die, S = {1, 2, 3, 4, 5, 6}, 1 1 2 1 6 for the roll

### Math 431 An Introduction to Probability. Final Exam Solutions

Math 43 An Introduction to Probability Final Eam Solutions. A continuous random variable X has cdf a for 0, F () = for 0 <

### University of California, Los Angeles Department of Statistics. Random variables

University of California, Los Angeles Department of Statistics Statistics Instructor: Nicolas Christou Random variables Discrete random variables. Continuous random variables. Discrete random variables.

### WHERE DOES THE 10% CONDITION COME FROM?

1 WHERE DOES THE 10% CONDITION COME FROM? The text has mentioned The 10% Condition (at least) twice so far: p. 407 Bernoulli trials must be independent. If that assumption is violated, it is still okay

### Chapter 5 Discrete Probability Distribution. Learning objectives

Chapter 5 Discrete Probability Distribution Slide 1 Learning objectives 1. Understand random variables and probability distributions. 1.1. Distinguish discrete and continuous random variables. 2. Able

### Homework 4 - KEY. Jeff Brenion. June 16, 2004. Note: Many problems can be solved in more than one way; we present only a single solution here.

Homework 4 - KEY Jeff Brenion June 16, 2004 Note: Many problems can be solved in more than one way; we present only a single solution here. 1 Problem 2-1 Since there can be anywhere from 0 to 4 aces, the

### Random variables P(X = 3) = P(X = 3) = 1 8, P(X = 1) = P(X = 1) = 3 8.

Random variables Remark on Notations 1. When X is a number chosen uniformly from a data set, What I call P(X = k) is called Freq[k, X] in the courseware. 2. When X is a random variable, what I call F ()

### MATH 140 Lab 4: Probability and the Standard Normal Distribution

MATH 140 Lab 4: Probability and the Standard Normal Distribution Problem 1. Flipping a Coin Problem In this problem, we want to simualte the process of flipping a fair coin 1000 times. Note that the outcomes

### Practice Exam 1: Long List 18.05, Spring 2014

Practice Eam : Long List 8.05, Spring 204 Counting and Probability. A full house in poker is a hand where three cards share one rank and two cards share another rank. How many ways are there to get a full-house?

### Minitab Guide. This packet contains: A Friendly Guide to Minitab. Minitab Step-By-Step

Minitab Guide This packet contains: A Friendly Guide to Minitab An introduction to Minitab; including basic Minitab functions, how to create sets of data, and how to create and edit graphs of different

### Chapter 4. Probability Distributions

Chapter 4 Probability Distributions Lesson 4-1/4-2 Random Variable Probability Distributions This chapter will deal the construction of probability distribution. By combining the methods of descriptive

### STT315 Chapter 4 Random Variables & Probability Distributions KM. Chapter 4.5, 6, 8 Probability Distributions for Continuous Random Variables

Chapter 4.5, 6, 8 Probability Distributions for Continuous Random Variables Discrete vs. continuous random variables Examples of continuous distributions o Uniform o Exponential o Normal Recall: A random

### Joint Exam 1/P Sample Exam 1

Joint Exam 1/P Sample Exam 1 Take this practice exam under strict exam conditions: Set a timer for 3 hours; Do not stop the timer for restroom breaks; Do not look at your notes. If you believe a question

### Chapter 5. Random variables

Random variables random variable numerical variable whose value is the outcome of some probabilistic experiment; we use uppercase letters, like X, to denote such a variable and lowercase letters, like

### Binomial Probability Distribution

Binomial Probability Distribution In a binomial setting, we can compute probabilities of certain outcomes. This used to be done with tables, but with graphing calculator technology, these problems are

### Question: What is the probability that a five-card poker hand contains a flush, that is, five cards of the same suit?

ECS20 Discrete Mathematics Quarter: Spring 2007 Instructor: John Steinberger Assistant: Sophie Engle (prepared by Sophie Engle) Homework 8 Hints Due Wednesday June 6 th 2007 Section 6.1 #16 What is the

### Normal distribution Approximating binomial distribution by normal 2.10 Central Limit Theorem

1.1.2 Normal distribution 1.1.3 Approimating binomial distribution by normal 2.1 Central Limit Theorem Prof. Tesler Math 283 October 22, 214 Prof. Tesler 1.1.2-3, 2.1 Normal distribution Math 283 / October

### PROBABILITIES AND PROBABILITY DISTRIBUTIONS

Published in "Random Walks in Biology", 1983, Princeton University Press PROBABILITIES AND PROBABILITY DISTRIBUTIONS Howard C. Berg Table of Contents PROBABILITIES PROBABILITY DISTRIBUTIONS THE BINOMIAL

### Chapter 4 & 5 practice set. The actual exam is not multiple choice nor does it contain like questions.

Chapter 4 & 5 practice set. The actual exam is not multiple choice nor does it contain like questions. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

### You flip a fair coin four times, what is the probability that you obtain three heads.

Handout 4: Binomial Distribution Reading Assignment: Chapter 5 In the previous handout, we looked at continuous random variables and calculating probabilities and percentiles for those type of variables.

### Math 141. Lecture 7: Variance, Covariance, and Sums. Albyn Jones 1. 1 Library 304. jones/courses/141

Math 141 Lecture 7: Variance, Covariance, and Sums Albyn Jones 1 1 Library 304 jones@reed.edu www.people.reed.edu/ jones/courses/141 Last Time Variance: expected squared deviation from the mean: Standard

### 4. Joint Distributions

Virtual Laboratories > 2. Distributions > 1 2 3 4 5 6 7 8 4. Joint Distributions Basic Theory As usual, we start with a random experiment with probability measure P on an underlying sample space. Suppose

### Ch5: Discrete Probability Distributions Section 5-1: Probability Distribution

Recall: Ch5: Discrete Probability Distributions Section 5-1: Probability Distribution A variable is a characteristic or attribute that can assume different values. o Various letters of the alphabet (e.g.

### Some special discrete probability distributions

University of California, Los Angeles Department of Statistics Statistics 100A Instructor: Nicolas Christou Some special discrete probability distributions Bernoulli random variable: It is a variable that

### Statistics 100A Homework 8 Solutions

Part : Chapter 7 Statistics A Homework 8 Solutions Ryan Rosario. A player throws a fair die and simultaneously flips a fair coin. If the coin lands heads, then she wins twice, and if tails, the one-half

### Chapter 3: DISCRETE RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS. Part 3: Discrete Uniform Distribution Binomial Distribution

Chapter 3: DISCRETE RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS Part 3: Discrete Uniform Distribution Binomial Distribution Sections 3-5, 3-6 Special discrete random variable distributions we will cover

### Normal distribution. ) 2 /2σ. 2π σ

Normal distribution The normal distribution is the most widely known and used of all distributions. Because the normal distribution approximates many natural phenomena so well, it has developed into a

### Lab 11. Simulations. The Concept

Lab 11 Simulations In this lab you ll learn how to create simulations to provide approximate answers to probability questions. We ll make use of a particular kind of structure, called a box model, that

### Math 425 (Fall 08) Solutions Midterm 2 November 6, 2008

Math 425 (Fall 8) Solutions Midterm 2 November 6, 28 (5 pts) Compute E[X] and Var[X] for i) X a random variable that takes the values, 2, 3 with probabilities.2,.5,.3; ii) X a random variable with the

### The normal approximation to the binomial

The normal approximation to the binomial The binomial probability function is not useful for calculating probabilities when the number of trials n is large, as it involves multiplying a potentially very

### Chapter 9 Monté Carlo Simulation

MGS 3100 Business Analysis Chapter 9 Monté Carlo What Is? A model/process used to duplicate or mimic the real system Types of Models Physical simulation Computer simulation When to Use (Computer) Models?

### Chapters 5. Multivariate Probability Distributions

Chapters 5. Multivariate Probability Distributions Random vectors are collection of random variables defined on the same sample space. Whenever a collection of random variables are mentioned, they are

### Toss a coin twice. Let Y denote the number of heads.

! Let S be a discrete sample space with the set of elementary events denoted by E = {e i, i = 1, 2, 3 }. A random variable is a function Y(e i ) that assigns a real value to each elementary event, e i.

### Chapter 5. Discrete Probability Distributions

Chapter 5. Discrete Probability Distributions Chapter Problem: Did Mendel s result from plant hybridization experiments contradicts his theory? 1. Mendel s theory says that when there are two inheritable

### 3 Multiple Discrete Random Variables

3 Multiple Discrete Random Variables 3.1 Joint densities Suppose we have a probability space (Ω, F,P) and now we have two discrete random variables X and Y on it. They have probability mass functions f

### P (x) 0. Discrete random variables Expected value. The expected value, mean or average of a random variable x is: xp (x) = v i P (v i )

Discrete random variables Probability mass function Given a discrete random variable X taking values in X = {v 1,..., v m }, its probability mass function P : X [0, 1] is defined as: P (v i ) = Pr[X =

### Statistics 100A Homework 4 Solutions

Problem 1 For a discrete random variable X, Statistics 100A Homework 4 Solutions Ryan Rosario Note that all of the problems below as you to prove the statement. We are proving the properties of epectation

### 6 PROBABILITY GENERATING FUNCTIONS

6 PROBABILITY GENERATING FUNCTIONS Certain derivations presented in this course have been somewhat heavy on algebra. For example, determining the expectation of the Binomial distribution (page 5.1 turned

GNH7/GEOLGG9/GEOL2 EARTHQUAKE SEISMOLOGY AND EARTHQUAKE HAZARD TUTORIAL (6): EARTHQUAKE STATISTICS Question. Questions and Answers How many distinct 5-card hands can be dealt from a standard 52-card deck?

Using Your TI-NSpire Calculator: Binomial Probability Distributions Dr. Laura Schultz Statistics I This handout describes how to use the binompdf and binomcdf commands to work with binomial probability

### Statistics 100A Homework 3 Solutions

Chapter Statistics 00A Homework Solutions Ryan Rosario. Two balls are chosen randomly from an urn containing 8 white, black, and orange balls. Suppose that we win \$ for each black ball selected and we

### RANDOM VARIABLES MATH CIRCLE (ADVANCED) 3/3/2013. 3 k) ( 52 3 )

RANDOM VARIABLES MATH CIRCLE (ADVANCED) //0 0) a) Suppose you flip a fair coin times. i) What is the probability you get 0 heads? ii) head? iii) heads? iv) heads? For = 0,,,, P ( Heads) = ( ) b) Suppose

### Notes on Continuous Random Variables

Notes on Continuous Random Variables Continuous random variables are random quantities that are measured on a continuous scale. They can usually take on any value over some interval, which distinguishes

### ST 371 (IV): Discrete Random Variables

ST 371 (IV): Discrete Random Variables 1 Random Variables A random variable (rv) is a function that is defined on the sample space of the experiment and that assigns a numerical variable to each possible

### Probability: Terminology and Examples Class 2, 18.05, Spring 2014 Jeremy Orloff and Jonathan Bloom

Probability: Terminology and Examples Class 2, 18.05, Spring 2014 Jeremy Orloff and Jonathan Bloom 1 Learning Goals 1. Know the definitions of sample space, event and probability function. 2. Be able to

### Introduction to the Practice of Statistics Fifth Edition Moore, McCabe

Introduction to the Practice of Statistics Fifth Edition Moore, McCabe Section 5.1 Homework Answers 5.7 In the proofreading setting if Exercise 5.3, what is the smallest number of misses m with P(X m)

### Math 202-0 Quizzes Winter 2009

Quiz : Basic Probability Ten Scrabble tiles are placed in a bag Four of the tiles have the letter printed on them, and there are two tiles each with the letters B, C and D on them (a) Suppose one tile

### Joint Probability Distributions and Random Samples (Devore Chapter Five)

Joint Probability Distributions and Random Samples (Devore Chapter Five) 1016-345-01 Probability and Statistics for Engineers Winter 2010-2011 Contents 1 Joint Probability Distributions 1 1.1 Two Discrete

### Normal Approximation. Contents. 1 Normal Approximation. 1.1 Introduction. Anthony Tanbakuchi Department of Mathematics Pima Community College

Introductory Statistics Lectures Normal Approimation To the binomial distribution Department of Mathematics Pima Community College Redistribution of this material is prohibited without written permission

### Probability & Statistics Primer Gregory J. Hakim University of Washington 2 January 2009 v2.0

Probability & Statistics Primer Gregory J. Hakim University of Washington 2 January 2009 v2.0 This primer provides an overview of basic concepts and definitions in probability and statistics. We shall

### Probability and Statistical Methods. Chapter 4 Mathematical Expectation

Math 3 Chapter 4 Mathematical Epectation Mean of a Random Variable Definition. Let be a random variable with probability distribution f( ). The mean or epected value of is, f( ) µ = µ = E =, if is a discrete

### The normal approximation to the binomial

The normal approximation to the binomial In order for a continuous distribution (like the normal) to be used to approximate a discrete one (like the binomial), a continuity correction should be used. There

### Overview of Monte Carlo Simulation, Probability Review and Introduction to Matlab

Monte Carlo Simulation: IEOR E4703 Fall 2004 c 2004 by Martin Haugh Overview of Monte Carlo Simulation, Probability Review and Introduction to Matlab 1 Overview of Monte Carlo Simulation 1.1 Why use simulation?

### Probability. A random sample is selected in such a way that every different sample of size n has an equal chance of selection.

1 3.1 Sample Spaces and Tree Diagrams Probability This section introduces terminology and some techniques which will eventually lead us to the basic concept of the probability of an event. The Rare Event

### 2. Discrete random variables

2. Discrete random variables Statistics and probability: 2-1 If the chance outcome of the experiment is a number, it is called a random variable. Discrete random variable: the possible outcomes can be

### ( ) is proportional to ( 10 + x)!2. Calculate the

PRACTICE EXAMINATION NUMBER 6. An insurance company eamines its pool of auto insurance customers and gathers the following information: i) All customers insure at least one car. ii) 64 of the customers

### Probability Distributions

Learning Objectives Probability Distributions Section 1: How Can We Summarize Possible Outcomes and Their Probabilities? 1. Random variable 2. Probability distributions for discrete random variables 3.

### STATISTICS FOR PSYCH MATH REVIEW GUIDE

STATISTICS FOR PSYCH MATH REVIEW GUIDE ORDER OF OPERATIONS Although remembering the order of operations as BEDMAS may seem simple, it is definitely worth reviewing in a new context such as statistics formulae.

### We have discussed the notion of probabilistic dependence above and indicated that dependence is

1 CHAPTER 7 Online Supplement Covariance and Correlation for Measuring Dependence We have discussed the notion of probabilistic dependence above and indicated that dependence is defined in terms of conditional

### 7.7 Solving Rational Equations

Section 7.7 Solving Rational Equations 7 7.7 Solving Rational Equations When simplifying comple fractions in the previous section, we saw that multiplying both numerator and denominator by the appropriate

### Chapter 7 Part 2. Hypothesis testing Power

Chapter 7 Part 2 Hypothesis testing Power November 6, 2008 All of the normal curves in this handout are sampling distributions Goal: To understand the process of hypothesis testing and the relationship

### Probability Calculator

Chapter 95 Introduction Most statisticians have a set of probability tables that they refer to in doing their statistical wor. This procedure provides you with a set of electronic statistical tables that

### Homework 6 (due November 4, 2009)

Homework 6 (due November 4, 2009 Problem 1. On average, how many independent games of poker are required until a preassigned player is dealt a straight? Here we define a straight to be cards of consecutive

### Probability and Statistics Vocabulary List (Definitions for Middle School Teachers)

Probability and Statistics Vocabulary List (Definitions for Middle School Teachers) B Bar graph a diagram representing the frequency distribution for nominal or discrete data. It consists of a sequence

### Mathematical Expectation

Mathematical Expectation Properties of Mathematical Expectation I The concept of mathematical expectation arose in connection with games of chance. In its simplest form, mathematical expectation is the

### Chapter 5 - Practice Problems 1

Chapter 5 - Practice Problems 1 Identify the given random variable as being discrete or continuous. 1) The number of oil spills occurring off the Alaskan coast 1) A) Continuous B) Discrete 2) The ph level

### ST 371 (VIII): Theory of Joint Distributions

ST 371 (VIII): Theory of Joint Distributions So far we have focused on probability distributions for single random variables. However, we are often interested in probability statements concerning two or

### P(X = x k ) = 1 = k=1

74 CHAPTER 6. IMPORTANT DISTRIBUTIONS AND DENSITIES 6.2 Problems 5.1.1 Which are modeled with a unifm distribution? (a Yes, P(X k 1/6 f k 1,...,6. (b No, this has a binomial distribution. (c Yes, P(X k

### Joint Probability Distributions and Random Samples. Week 5, 2011 Stat 4570/5570 Material from Devore s book (Ed 8), and Cengage

5 Joint Probability Distributions and Random Samples Week 5, 2011 Stat 4570/5570 Material from Devore s book (Ed 8), and Cengage Two Discrete Random Variables The probability mass function (pmf) of a single

### STAT 35A HW2 Solutions

STAT 35A HW2 Solutions http://www.stat.ucla.edu/~dinov/courses_students.dir/09/spring/stat35.dir 1. A computer consulting firm presently has bids out on three projects. Let A i = { awarded project i },

### Examination 110 Probability and Statistics Examination

Examination 0 Probability and Statistics Examination Sample Examination Questions The Probability and Statistics Examination consists of 5 multiple-choice test questions. The test is a three-hour examination

### 10.1. Solving Quadratic Equations. Investigation: Rocket Science CONDENSED

CONDENSED L E S S O N 10.1 Solving Quadratic Equations In this lesson you will look at quadratic functions that model projectile motion use tables and graphs to approimate solutions to quadratic equations

### Lecture 6: Discrete & Continuous Probability and Random Variables

Lecture 6: Discrete & Continuous Probability and Random Variables D. Alex Hughes Math Camp September 17, 2015 D. Alex Hughes (Math Camp) Lecture 6: Discrete & Continuous Probability and Random September

### Department of Mathematics, Indian Institute of Technology, Kharagpur Assignment 2-3, Probability and Statistics, March 2015. Due:-March 25, 2015.

Department of Mathematics, Indian Institute of Technology, Kharagpur Assignment -3, Probability and Statistics, March 05. Due:-March 5, 05.. Show that the function 0 for x < x+ F (x) = 4 for x < for x

### MBA 611 STATISTICS AND QUANTITATIVE METHODS

MBA 611 STATISTICS AND QUANTITATIVE METHODS Part I. Review of Basic Statistics (Chapters 1-11) A. Introduction (Chapter 1) Uncertainty: Decisions are often based on incomplete information from uncertain

### Section 6.1 Discrete Random variables Probability Distribution

Section 6.1 Discrete Random variables Probability Distribution Definitions a) Random variable is a variable whose values are determined by chance. b) Discrete Probability distribution consists of the values

### An approach to Calculus of Probabilities through real situations

MaMaEuSch Management Mathematics for European Schools http://www.mathematik.unikl.de/ mamaeusch An approach to Calculus of Probabilities through real situations Paula Lagares Barreiro Federico Perea Rojas-Marcos

### MA 1125 Lecture 14 - Expected Values. Friday, February 28, 2014. Objectives: Introduce expected values.

MA 5 Lecture 4 - Expected Values Friday, February 2, 24. Objectives: Introduce expected values.. Means, Variances, and Standard Deviations of Probability Distributions Two classes ago, we computed the

### 2 Binomial, Poisson, Normal Distribution

2 Binomial, Poisson, Normal Distribution Binomial Distribution ): We are interested in the number of times an event A occurs in n independent trials. In each trial the event A has the same probability

### Chapter 6 Random Variables

Chapter 6 Random Variables Day 1: 6.1 Discrete Random Variables Read 340-344 What is a random variable? Give some examples. A numerical variable that describes the outcomes of a chance process. Examples:

### 6. Jointly Distributed Random Variables

6. Jointly Distributed Random Variables We are often interested in the relationship between two or more random variables. Example: A randomly chosen person may be a smoker and/or may get cancer. Definition.

### 4.1 4.2 Probability Distribution for Discrete Random Variables

4.1 4.2 Probability Distribution for Discrete Random Variables Key concepts: discrete random variable, probability distribution, expected value, variance, and standard deviation of a discrete random variable.

### MAS108 Probability I

1 QUEEN MARY UNIVERSITY OF LONDON 2:30 pm, Thursday 3 May, 2007 Duration: 2 hours MAS108 Probability I Do not start reading the question paper until you are instructed to by the invigilators. The paper

### Random Variables. Chapter 2. Random Variables 1

Random Variables Chapter 2 Random Variables 1 Roulette and Random Variables A Roulette wheel has 38 pockets. 18 of them are red and 18 are black; these are numbered from 1 to 36. The two remaining pockets

### SUNY ECC. ACCUPLACER Preparation Workshop. Algebra Skills

SUNY ECC ACCUPLACER Preparation Workshop Algebra Skills Gail A. Butler Ph.D. Evaluating Algebraic Epressions Substitute the value (#) in place of the letter (variable). Follow order of operations!!! E)

### Probability and Statistics

CHAPTER 2: RANDOM VARIABLES AND ASSOCIATED FUNCTIONS 2b - 0 Probability and Statistics Kristel Van Steen, PhD 2 Montefiore Institute - Systems and Modeling GIGA - Bioinformatics ULg kristel.vansteen@ulg.ac.be

### Discrete Random Variables and their Probability Distributions

CHAPTER 5 Discrete Random Variables and their Probability Distributions CHAPTER OUTLINE 5.1 Probability Distribution of a Discrete Random Variable 5.2 Mean and Standard Deviation of a Discrete Random Variable

### Section 5 Part 2. Probability Distributions for Discrete Random Variables

Section 5 Part 2 Probability Distributions for Discrete Random Variables Review and Overview So far we ve covered the following probability and probability distribution topics Probability rules Probability

### Solution. Solution. (a) Sum of probabilities = 1 (Verify) (b) (see graph) Chapter 4 (Sections 4.3-4.4) Homework Solutions. Section 4.

Math 115 N. Psomas Chapter 4 (Sections 4.3-4.4) Homework s Section 4.3 4.53 Discrete or continuous. In each of the following situations decide if the random variable is discrete or continuous and give