MT426 Notebook 3 Fall 2012 prepared by Professor Jenny Baglivo. 3 MT426 Notebook Definitions Joint Discrete Distributions...

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1 MT426 Notebook 3 Fall 2012 prepared by Professor Jenny Baglivo c Copyright by Jenny A. Baglivo. All Rights Reserved. Contents 3 MT426 Notebook Definitions Joint Discrete Distributions Bivariate Distribution, PDF, CDF, Marginal Distributions Conditional Distributions, Regression Independent Random Variables Bivariate Hypergeometric Distribution Trinomial Distribution Simple Random Sample, Trinomial Approximation, Survey Analysis Joint Continuous Distributions Bivariate Distribution, PDF, CDF, Marginal Distributions Conditional Distributions, Regression Independent Random Variables Bivariate Uniform Distribution Bivariate Normal Distribution Transforming Random Variables, CDF Method Multivariate Distributions, Random Samples Multivariate Hypergeometric and Multinomial Distributions Mutual Independence, Repeated Trials, Random Samples

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3 3 MT426 Notebook 3 This notebook is concerned with joint discrete and joint continuous distributions. The notes correspond to material in Chapter 3 of the Rice textbook. 3.1 Definitions A probability distribution describing the joint variability of two or more random variables is called a joint distribution. A bivariate distribution is the joint distribution of a pair of random variables. Here are examples of situations we might want to study: 1. An urn contains 10 red chips, 8 blue chips, and 15 green chips. Let X be the number of red chips and Y be the number of blue chips in a subset of size 5 chosen from the urn. We wish to describe the joint distribution of the random pair (X, Y ). 2. Let X be the height (in feet), Y be the weight (in pounds), and Z be the serum cholesterol level (in mg/dl) of a person chosen from a given population. We wish to describe the joint distribution of the random triple (X, Y, Z). 3. A manufacturing process produces rods whose lengths (in inches) vary in the interval [0.98, 1.02]. We repeat the experiment choose a rod from those produced that day and measure its length on 8 separate occasions. We wish to describe the joint distribution of the random eight-tuple (X 1, X 2, X 3, X 4, X 5, X 6, X 7, X 8 ). 3.2 Joint Discrete Distributions A joint discrete distribution describes the joint variability of two or more discrete random variables Bivariate Distribution, PDF, CDF, Marginal Distributions Assume that X and Y are discrete random variables. The joint probability density function (joint PDF) (or joint frequency function (joint FF)) of the random pair (X, Y ) is defined as follows: p(x, y) = P (X = x, Y = y) for all real pairs (x, y), where the comma is understood to mean the intersection of events. The joint cumulative distribution function (joint CDF) of the random pair (X, Y ) is defined as follows: F (x, y) = P (X x, Y y) for all real pairs (x, y), where the comma is understood to mean the intersection of events. 3

4 If X and Y are discrete random variables with joint PDF p(x, y), then The marginal probability density function (marginal PDF) (or marginal frequency function (marginal FF)) of X is p X (x) = where R Y is the range of Y. y R Y p(x, y), for all real numbers x, The marginal probability density function (marginal PDF) (or marginal frequency function (marginal FF)) of Y is p Y (y) = where R X is the range of X. x R X p(x, y), for all real numbers y, Exercise 1 (Olkin et al, Macmillan 1994, page 545). In an experiment to study the relationship between spatial perception and the ability to use a graphical method to solve algebra problems, subjects were asked to solve 3 puzzles and 3 algebra problems. Let X be the number of puzzles correctly solved, and let Y be the number of algebra problems correctly solved. The following table summarizes the results, using proportions of individuals in each crossclassification: y = 0 y = 1 y = 2 y = 3 sum: x = x = x = x = sum: Consider the experiment Choose a name from the population of subjects and record (X, Y ) = (# puzzles solved, # algebra problems solved). If each choice is equally likely, then the body of the table gives the joint (X, Y ) distribution, the rightmost column of the table gives the marginal X distribution, the bottom row of the table gives the marginal Y distribution, and the plot on the right gives the joint probability histogram of the random pair. The joint probability histogram is constructed using a box whose base is a square of area 1 centered at (x, y), and whose height is p(x, y), for each pair with nonzero probability. The sum of the volumes of the boxes is 1. 4

5 Assume that the table represents the joint (X, Y ) distribution. (a) Find the probability that the subject (1) solved the same number of puzzles and algebra problems, P (X = Y ). (2) solved fewer puzzles than algebra problems, P (X < Y ). (3) solved more puzzles than algebra problems, P (X > Y ). (b) Let W = X + Y be the total number of problems solved by the subject. Completely specify the PDF of W. 5

6 Exercise 2. An urn contains 4 red chips, 3 white chips, and 1 blue chip. Consider the following 2-step experiment: Step 1: Step 2: Thoroughly mix the contents of the urn. Choose a chip and record its color. Return the chip plus two more chips of the same color to the urn. Thoroughly mix the contents of the urn. Choose a chip and record its color. Let X be the number of red chips, and Y be the number of white chips, chosen. Construct a table showing the joint (X, Y ) distribution, and the marginal X and Y distributions. 6

7 3.2.2 Conditional Distributions, Regression Let X and Y be discrete random variables. If P (Y = y) 0, then the conditional probability density function (conditional PDF) (or conditional frequency function (conditional FF)) of X given Y = y is defined as follows: p X Y =y (x y) = P (X = x Y = y) = P (X = x, Y = y) P (Y = y) for all real numbers x. If P (X = x) 0, then the conditional probability density function (conditional PDF) (or conditional frequency function (conditional FF)) of Y given X = x is defined as follows: p Y X=x (y x) = P (Y = y X = x) = P (X = x, Y = y) P (X = x) for all real numbers y. Note that in the first case, the conditional sample space is the collection of outcomes with Y = y; in the second case, it is the collection of outcomes with X = x. Exercise 1, continued. Consider again the joint puzzle-algebra problem distribution. (c) Use the information in the joint (X, Y ) distribution table to fill-in the following table with the conditional distribution of Y given X = 1: y p Y X=1 (y 1) (d) Use the information in the joint (X, Y ) distribution table to fill-in the following table with the conditional distribution of Y given X = 3: y p Y X=3 (y 3) 7

8 Here are probability histograms for the conditional distributions of Y given X = x, for each x: Y X = 0 Y X = 1 Y X = 2 Y X = 3 As x increases, the conditional distribution of Y given X = x changes, with more weight being given to larger values of Y. Regression. Conditional distributions are often used as weights in weighted averages. For example, suppose that we would like to find the average number of algebra problems solved given that a subject solved exactly x puzzles correctly, for each possible x. Then we would calculate a quantity known as the conditional mean (or conditional expectation) of Y given X = x, E(Y X = x) = 3 y p Y X=x (y x), x = 0, 1, 2, 3. y=0 The pairs (x, E(Y X = x)) are shown in the plot on the right. Note that as x increases, so does the average value of Y given X = x. The collection of conditional means E(Y X = x) is often called the regression of Y on X. 8

9 3.2.3 Independent Random Variables The discrete random variables X and Y are said to be independent if p(x, y) = p X (x)p Y (y) for all real pairs (x, y), where p(x, y) is the joint PDF, and p X (x) and p Y (y) are the marginal PDFs, and they are said to be dependent if p(x, y) p Y (x)p Y (y) for at least one real pair (x, y). Notes: 1. The definition says that the discrete random variables X and Y are independent if the probability of the intersection of events X = x and Y = y is equal to the product of the probabilities of the events for all possible x and y: P (X = x, Y = y) = P (X = x)p (Y = y) for all real pairs (x, y). 2. An equivalent definition of independence uses the cumulative distribution functions rather than the frequency functions. That is, X and Y are independent if F (x, y) = F X (x)f Y (y) for all real pairs (x, y), where F (x, y) is the joint CDF, and F X (x) and F Y (y) are the marginal CDFs, and they are said to be dependent if F (x, y) F X (x)f Y (y) for at least one real pair (x, y). 3. Sometimes the context of a problem implies independence. For example, suppose you roll two fair six-sided dice (one red and one blue) and let X be the number on the top face of the red die and Y be the number on the top face of the blue die. Then, the context of the problem tells us that X and Y are independent random variables. 4. If X and Y are independent, then conditional and marginal distributions are equal. To see this (please complete): 9

10 3.2.4 Bivariate Hypergeometric Distribution Let n, M 1, M 2, and M 3 be positive integers with n < M 1 + M 2 + M 3. Then (X, Y ) is said to have a bivariate hypergeometric distribution with parameters n and (M 1, M 2, M 3 ) if its joint PDF has the following form: ) ( M2 ) ( M3 ) p(x, y) = P (X = x, Y = y) = when x and y are non-negative integers satisfying ( M1 x n x y y ( M1 +M 2 +M 3 n x min(n, M 1 ), y min(n, M 2 ), and max(0, n M 3 ) x + y min(n, M 3 ), and is equal to zero otherwise. Model for urn experiments. Bivariate hypergeometric distributions are used to model urn experiments, where the urn contains N = M 1 + M 2 + M 3 objects, with M i of type i for i = 1, 2, 3. Let X equal the number of objects of type 1 and Y equal the number of objects of type 2 in a subset of size n chosen from the urn. If each choice of subset is equally likely, then (X, Y ) has a bivariate hypergeometric distribution with parameters n and (M 1, M 2, M 3 ). Marginal and conditional distributions. If (X, Y ) has a bivariate hypergeometric distribution, then X has a hypergeometric distribution with parameters n, M 1, and N, and Y has a hypergeometric distribution with parameters n, M 2, and N. In addition, each conditional distribution is hypergeometric. ) Exercise. An urn contains 5 red, 3 white, and 4 blue balls. Let X be the number of red balls and Y be the number of white balls in a subset of size 4. Assume each choice of subset is equally likely. (a) Write the formula for the joint PDF of (X, Y ), with appropriate ranges. 10

11 (b) The table on the right gives the numerators in the formula for joint probabilities. Use this table to find the following probabilities: y = 0 y = 1 y = 2 y = 3 sum: x = x = x = x = x = sum: P (X < 2, Y < 2) P (X < 2, Y 2) P (X 2, Y < 2) P (X 2, Y 2) (c) X has a hypergeometric distribution based on choosing a subset of size containing with a total of from an urn special objects, objects. (d) Y given X = 1 has a hypergeometric distribution based on choosing a subset of size containing with a total of from an urn special objects, objects. (e) Are X and Y independent? Why? 11

12 3.2.5 Trinomial Distribution Let n be a positive integer, and p 1, p 2, and p 3 be positive proportions with sum 1. Then (X, Y ) is said to have a trinomial distribution with parameters n and (p 1, p 2, p 3 ) when its joint PDF has the following form: p(x, y) = P (X = x, Y = y) = ( n ) x, y, n x y p x 1 p y 2 pn x y 3, when x = 0, 1,..., n; y = 0, 1,..., n; x + y n, and is equal to zero otherwise. Model for counts of independent trials. Trinomial distributions can be used to model counts related to independent trials of an experiment with exactly three outcomes. Specifically, suppose that an experiment has three outcomes which occur with probabilities p 1, p 2, and p 3, respectively. Let X be the number of occurrences of outcome 1 and Y be the number of occurrences of outcome 2 in n independent trials of the experiment. Then the random pair (X, Y ) has a trinomial distribution with parameters n and (p 1, p 2, p 3 ). Marginal and conditional distributions. If (X, Y ) has a trinomial distribution with parameters n and (p 1, p 2, p 3 ), then X has a binomial distribution with parameters n and p 1, and Y has a binomial distribution with parameters n and p 2. In addition, each conditional distribution is binomial. Exercise. An urn contains 5 red, 3 white, and 4 blue balls. You repeat the following experiment 4 times: Thoroughly mix the contents of the urn, choose one ball, note its color (R, W, B), return the ball to the urn. Let X be the number of R s and let Y be the number of W s in the list of four colors. (a) Write the formula for the joint PDF of (X, Y ), with appropriate ranges. 12

13 (b) The table on the right gives the joint and marginal distributions, rounded to three decimal places of accuracy. Use this table to find the following probabilities: y = 0 y = 1 y = 2 y = 3 y = 4 sum: x = x = x = x = x = sum: P (X < 2, Y < 2) P (X < 2, Y 2) P (X 2, Y < 2) P (X 2, Y 2) (c) X has a binomial distribution based on independent trials of an experiment with success probability. (d) Y given X = 1 has a binomial distribution based on independent trials of an experiment with success probability. (e) Are X and Y independent? Why? 13

14 3.2.6 Simple Random Sample, Trinomial Approximation, Survey Analysis Suppose that an urn contains N objects. A simple random sample of size n is a sequence of n objects chosen without replacement from the urn, where the choice of each sequence is equally likely. Let M i be the number of objects of type i in the urn, for i = 1, 2, 3, X be the number of objects of type 1, and Y be the number of objects of type 2, in a simple random sample of size n. Then the random pair (X, Y ) has a bivariate hypergeometric distribution with parameters n and (M 1, M 2, M 3 ). Further, if N is large, then trinomial probabilities can be used to approximate bivariate hypergeometric probabilities: Theorem (Trinomial Approximation). If N is large and p i = M i /N is not too extreme, for i = 1, 2, 3, then the trinomial distribution with parameters n and (p 1, p 2, p 3 ) can be used to approximate the bivariate hypergeometric distribution with parameters n and (M 1, M 2, M 3 ). Specifically, P (x of type 1 and y of type 2) = ( M1 x )( M2 )( M3 y n x y for each (x, y) within the appropriate range. ) ( ( N n) n x, y, n x y ) p x 1p y 2 pn x y 3 The theorem tells us that sampling with replacement can be used to approximate sampling without replacement. The approximation is good when n <.01N and each p i = M i N (.05,.95). Survey analysis. Simple random samples are used in surveys. If the survey population is small, then bivariate hypergeometric distributions are used to analyze the results. If the survey population is large, then trinomial distributions are used to analyze the results, even though each person s opinion is solicited at most once. For example, suppose that a surveyor is interested in determining the level of support for a proposal to change the local tax structure, and decides to choose a simple random sample of size 10 from the registered voter list. If there are a total of 120 registered voters, where one-third support the proposal, one-half oppose the proposal, and one-sixth have no opinion, then the probability that exactly 3 support, 5 oppose, and 2 have no opinion is ( 40 ) ( 60 ) ( 20 ) P (X = 3, Y = 5) = ( 120 ) If there are thousands of registered voters, then the probability is ( ) ( ) ( ) 1 5 ( ) 1 2 P (X = 3, Y = 5) , 5, Note that in the approximation you do not need to know the exact number of registered voters. 14

15 3.3 Joint Continuous Distributions A joint continuous distribution describes the joint variability of two or more continuous random variables Bivariate Distribution, PDF, CDF, Marginal Distributions Assume that X and Y are continuous random variables. The joint cumulative distribution function (joint CDF) of the random pair (X, Y ) is defined as follows: F (x, y) = P (X x, Y y) for all real pairs (x, y), where the comma is understood to mean the intersection of events. If F (x, y) has continuous second partial derivatives, then the joint probability density function (joint PDF) of the random pair (X, Y ) is defined as follows: f(x, y) = for all possible pairs (x, y). 2 F (x, y) = 2 x y F (x, y), y x Computing probabilities using the joint PDF. If R is the joint range of (X, Y ) and D R 2, then the probability of the event the value of (X, Y ) is in the domain D is obtained by finding the volume under the density surface z = f(x, y) for (x, y) D R: P ((X, Y ) D) = f da. D R Marginal distributions. f(x, y), then If X and Y are continuous random variables with joint PDF The marginal probability density function (marginal PDF) of X is f X (x) = f(x, y) dy, for all real numbers x, R Y where R Y is the range of Y. The marginal probability density function (marginal PDF) of Y is f Y (y) = f(x, y) dx, for all real numbers y, R X where R X is the range of X. 15

16 Exercise 1. Let (X, Y ) be the random pair with joint density function f(x, y) = 1 8 (x2 + y 2 ), for (x, y) [ 1, 2] [ 1, 1], and 0 otherwise. (a) Find P (X > Y ). 16

17 (b) Completely specify the marginal PDF of X. (c) Completely specify the marginal PDF of Y. 17

18 Exercise 2. Let (X, Y ) be the random pair with joint density function f(x, y) = 1 4 e y/2, for 0 x y, and 0 otherwise. (a) Find P (X + Y < 5). 18

19 (b) Completely specify the marginal PDF of X. (c) Completely specify the marginal PDF of Y. 19

20 3.3.2 Conditional Distributions, Regression Let X and Y be continuous random variables with joint PDF f(x, y) and marginal PDFs f X (x) and f Y (y), respectively. If f Y (y) 0, then the conditional probability density function (conditional PDF) of X given Y = y is defined as follows: f X Y =y (x y) = f(x, y) f Y (y) for all real numbers x. If f X (x) 0, then the conditional probability density function (conditional PDF) of Y given X = x is defined as follows: f Y X=x (y x) = f(x, y) f X (x) for all real numbers y. In each case, the marginal PDF value is used to re-scale a one-variable function so that it becomes a valid density function. Exercise 1, continued. Let (X, Y ) be the continuous random pair whose joint PDF is f(x, y) = 1 8 (x2 + y 2 ), for (x, y) [ 1, 2] [ 1, 1], and 0 otherwise. Completely specify: (1) The conditional PDF of X given Y = y. (2) The conditional PDF of Y given X = x. 20

21 Exercise 2, continued. Let (X, Y ) be the continuous random pair whose joint PDF is f(x, y) = 1 4 e y/2, for 0 x y, and 0 otherwise. Completely specify: (1) The conditional PDF of X given Y = y. (2) The conditional PDF of Y given X = x. Regression. Conditional distributions are often used as weights in weighted averages. For example, suppose that we would like to find the average value of Y for each possible x in the problem above. Then we would calculate a quantity known as the conditional mean (or the conditional expectation) of Y given X = x, E(Y X = x) = x y f Y X=x (y x) dy. The pairs (x, E(Y X = x)) are shown in the plot on the right. Note that as x increases, so does the average value of Y given X = x. The collection of conditional means E(Y X = x) is often called the regression of Y on X. 21

22 3.3.3 Independent Random Variables The continuous random variables X and Y are said to be independent if F (x, y) = F X (x)f Y (y) for all real pairs (x, y), where F (x, y) is the joint CDF, and F X (x) and F Y (y) are the marginal CDFs, and they are said to be dependent if F (x, y) F X (x)f Y (y) for at least one real pair (x, y). Notes: 1. If F (x, y) has continuous second partial derivatives, then we can write an equivalent definition in terms of the density functions. That is, the continuous random variables X and Y are said to be independent if f(x, y) = f X (x)f Y (y) for all real pairs (x, y), where f(x, y) is the joint PDF, and f X (x) and f Y (y) are the marginal PDFs, and they are said to be dependent if f(x, y) f X (x)f Y (y) for at least one real pair (x, y). 2. Sometimes the context of a problem implies independence. For example, let X be the height of a randomly chosen woman living in the United States and Y be the height of a randomly chosen man living in the United States. Then, the context of the problem tells us that X and Y are independent random variables. 3. If X and Y are independent, then conditional and marginal distributions are equal. To see this (please complete), 22

23 Exercise. Let X and Y be independent continuous random variables. Let 1. D 1 R X be a subset of the range of nonzero density of X and 2. D 2 R Y be a subset of the range of nonzero density of Y. Demonstrate that P ((X, Y ) D 1 D 2 ) = P (X D 1 )P (Y D 2 ). 23

24 3.3.4 Bivariate Uniform Distribution Let R R 2 be a region of finite positive area. The random pair (X, Y ) is said to have a bivariate uniform distribution on R when its joint PDF has the following form: f(x, y) = 1 Area(R), for (x, y) R, and 0 otherwise. Exercise. Let (X, Y ) have a bivariate uniform distribution on the region R. Demonstrate that the probability that (X, Y ) lies in a subregion of R depends only on the area of the subregion. Exercise. Let (X, Y ) be the random pair with joint density function f(x, y) = 1 π, when x2 + y 2 1, and 0 otherwise. (a) Use geometry to find P (X > Y ). 24

25 (b) Use geometry to find P (X > 1/2). (c) Completely specify the marginal PDF of X. 25

26 Exercise. Let (X, Y ) have a bivariate uniform distribution on R = [5, 10] [0, 10]. (a) Find P (X < 2Y ). (b) Completely specify the marginal PDFs of X and Y. (c) Are X and Y independent? Why? 26

27 3.3.5 Bivariate Normal Distribution Let µ x and µ y be real numbers, σ x and σ y be positive real numbers, and ρ be a number in the interval 1 < ρ < 1. The random pair (X, Y ) is said to have a bivariate normal distribution with parameters µ x, µ y, σ x, σ y and ρ when its joint PDF is ( ) 1 term f(x, y) = 2πσ x σ exp y 1 ρ 2 2(1 ρ 2 for all real pairs (x, y) ) where ( ) x 2 ( ) ( ) ( µx x µx y µy y µy term = 2ρ + σ x and exp() is the exponential function. σ x σ y σ y ) 2 Note that X is a normal random variable with mean µ x and standard deviation σ x, and Y is a normal random variable with mean µ y and standard deviation σ y. Correlation coefficient. The parameter ρ is called the correlation coefficient. The correlation coefficient is a measure of the strength of the association between X and Y. Standard bivariate normal distribution. If µ x = µ y = 0 and σ x = σ y = 1, then the random pair (X, Y ) is said to have a standard bivariate normal distribution with parameter ρ. The joint density is as follows: ( 1 (x 2 f(x, y) = 2π 1 ρ exp 2ρxy + y 2 ) ) 2 2(1 ρ 2 for all real pairs (x, y). ) In this case, both X and Y are standard normal random variables. The following plots show the shape of the density surface for three different values of the correlation coefficient: ρ = 7 10 ρ = 0 ρ =

28 3.3.6 Transforming Random Variables, CDF Method Let (X, Y ) be a continuous random pair and let W = g(x, Y ), where g is a continuous function. Our goal is to determine the W distribution. The first steps will be to compute P (W w) = P (g(x, Y ) w) as an expression in w, and to find d dw P (W w), for each w in the range of W. Exercise. Let (X, Y ) have a bivariate uniform distribution on the disk R = {(x, y) x 2 + y 2 1}, and let W = X 2 + Y 2 be the distance of a random point to the origin. (Curves with distance w = 0.25, 0.50, 0.75, 1.00 are shown in the plot.) Completely specify the PDF of W. 28

29 Exercise. Let X and Y be the lengths of adjacent sides of a rectangle, and let W = XY be the area of the rectangle. Assume that X has a uniform distribution on the interval (0, 2), Y has a uniform distribution on the interval (0, 1), and that X and Y are independent. (Curves for areas w = 0.2, 0.6, 1.0, 1.4 are shown in the plot.) Completely specify the PDF of W. 29

30 Exercise. Assume that (X, Y ) has the joint PDF f(x, y) = 1 4 e y/2 when 0 < x < y, and 0 otherwise, and let W be the ratio W = Y/X. (Curves with ratios w = 2, 3, 4, 5 are shown in the plot.) Completely specify the PDF of W. 30

31 Exercise. Assume that (X, Y ) has the joint PDF f(x, y) = 1 x 2 y 2 when x > 1, y > 1, and 0 otherwise, and let W be the product W = Y X. (Curves with products w = 2, 3,..., 9 are shown in the plot.) Completely specify the PDF of W. 31

32 3.4 Multivariate Distributions, Random Samples A multivariate distribution is the joint distribution of k random variables, X 1, X 2,..., X k. Ideas studied in the bivariate case (k = 2) can be generalized to the case where k > 2. Joint CDF: The joint CDF of the random k-tuple (X 1, X 2,..., X k ) is defined as follows: F (x 1, x 2,..., x k ) = P (X 1 x 1, X 2 x 2,..., X k x k ) for all (x 1, x 2,..., x k ) R k, where commas are understood to mean the intersection of events. Joint PDF: The definition of the joint PDF depends on the types of random variables: 1. Discrete case: If the X i s are discrete, their joint PDF is defined as follows: p(x 1, x 2,..., x k ) = P (X 1 = x 1, X 2 = x 2,..., X k = x k ) for all (x 1, x 2,..., x k ) R k, where commas are understood to mean intersection. 2. Continuous case: If the X i s are continuous and F (x 1, x 2,..., x k ) has continuous k th order partial derivatives, then the joint PDF is defined as follows: f(x 1, x 2,..., x k ) = for all possible (x 1, x 2,..., x k ). k x 1 x k F (x 1, x 2,..., x k ) Multivariate Hypergeometric and Multinomial Distributions The following two multivariate families generalize the bivariate families we studied earlier: Multivariate Hypergeometric Distribution: Let n and M i, for i = 1,..., k, be positive integers with n < M 1 + M M k. The random k-tuple (X 1, X 2,..., X k ) is said to have a multivariate hypergeometric distribution with parameters n and (M 1, M 2,..., M k ) when its joint PDF has the following form: p(x 1, x 2,..., x k ) = ( M1 )( M2 ) ( x 1 x 2 Mk ( N, n) x k ) where N = M 1 + M M k, each x i is an integer satisfying 0 x i min(n, M i ), and the sum of the x i s is exactly n; otherwise, the joint PDF equals 0. Urn Sample Type 1 M 1 X 1 Type 2 M 2 X 2... Type k M k X k N n 32

33 Multinomial Distribution: Notes: Let n be a positive integer, and p i, for i = 1, 2,..., k, be positive proportions whose sum is exactly 1. The random k-tuple (X 1, X 2,..., X k ) is said to have a multinomial distribution with parameters n and (p 1, p 2,..., p k ) when its joint PDF has the following form: ( ) n p(x 1, x 2,..., x k ) = p x 1 1 x 1, x 2,..., x px 2 2 px k k, k when each x i is an integer satisfying 0 x i n, and the sum of the x i s is exactly n; otherwise, the joint PDF equals 0. Probs. Sample Type 1 p 1 X 1 Type 2 p 2 X 2... Type k p k X k 1 n 1. The multivariate hypergeometric distribution is used, for example, to model experiments where sampling is done without replacement from an urn containing exacty k types of objects, and the multinomial distribution is used to model experiments where sampling is done with replacement. 2. If N is large enough and each p i = M i /N is not too extreme, then the multivariate hypergeometric distribution is well-approximated by a multinomial distribution where p i = M i N for each i. (Thus, sampling with replacement can be used to approximate sampling without replacement.) Mutual Independence, Repeated Trials, Random Samples X 1, X 2,..., X k are said to be mutually independent (or independent when the context is clear) if F (x 1, x 2,..., x k ) = F 1 (x 1 )F 2 (x 2 ) F k (x k ) for all (x 1, x 2,..., x k ) R k, where F i (x i ) = P (X i x i ) for i = 1, 2,..., k. 1. Discrete case: Equivalently, the discrete X i s are mutually independent if p(x 1, x 2,..., x k ) = p 1 (x 1 )p 2 (x 2 ) p k (x k ) for all (x 1, x 2,..., x k ) R k, where p i (x i ) = P (X i = x i ) for i = 1, 2,..., k. 2. Continuous case: If the joint PDF exists, then the continuous X i s are mutually independent if f(x 1, x 2,..., x k ) = f 1 (x 1 )f 2 (x 2 ) f k (x k ) for all possible (x 1, x 2,..., x k ), where f i (x i ) is the density function of X i, for i = 1, 2..., k. 33

34 Exercise. Suppose that an urn contains 4 slips of paper with the numbers 1, 2, 3 and 4 written on them. The urn is thoroughly mixed, one slip of paper is removed, and the number is recorded. Let Let X = 1 if slip 1 or 2 is drawn, and let X = 0 otherwise. Let Y = 1 if slip 1 or 3 is drawn, and let Y = 0 otherwise. Let Z = 1 if slip 1 or 4 is drawn, and let Z = 0 otherwise. (a) Are X and Y independent? Are X and Z independent? Are Y and Z independent? (b) Are X, Y and Z mutually independent? 34

35 Mutual independence and probabilities: random variables, then If X 1, X 2,..., X k are mutually independent P ((X 1,..., X k ) D 1 D 2 D k ) = P (X 1 D 1 )P (X 2 D 2 ) P (X k D k ). Random samples: If X 1, X 2,..., X k are mutually independent and have a common distribution (each marginal distribution is the same), then X 1, X 2,..., X k are said to be a random sample from that distribution. Repeated trials and random samples: Consider k repetitions of an experiment, with the outcomes of the trials having no influence on one another, and let X i be the result of the i th repetition, for i = 1, 2,..., k. Then the X i s are mutually independent, and form a random sample from the common distribution. Exercise. Let X be the length in inches of a rod produced by a manufacturing process. Assume that X has a uniform distribution on the interval [0.98, 1.02]. On five separate days, a quality control engineer chooses one rod randomly from those produced that day and measures its length. Find the probability that all 5 lengths are greater than 0.99 inches long. 35

36 Exercise. Let X 1, X 2, X 3 be a random sample from the distribution with PDF x p(x) Let Y = max(x 1, X 2, X 3 ) be the maximum of the three random numbers. (a) Find P (Y y), for y = 1, 2, 3, 4. (b) Use your answer to part (a) to find P (Y = y) for y = 1, 2, 3, 4. 36

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