Statistics for Economists Lectures 6 & 7. Asrat Temesgen Stockholm University

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1 Statistics for Economists Lectures 6 & 7 Asrat Temesgen Stockholm University 1

2 Chapter 4- Bivariate Distributions 41 Distributions of two random variables Definition 41-1: Let X and Y be two random variables defined on a discrete probability space Let S denote the corresponding two-dimensional space of X and Y, the two random variables of the discrete type The probability that X= and Y= is denoted by f(, ) = P(X=, Y= ) The function f(, ) is called the joint probability mass function (joint pmf) of X and Y and has the following properties: a) b) where A is a subset of the space S Definition 41-2: Let X and Y have the joint Pmf f(x,y) with space S The probability mass function of X alone, which is called the marginal probability mass function of X, is defined by, where the summation is taken over all possible values for each given x in the x space S1 Similarly, the marginal pmf of Y is defined by where the summation is taken over all possible x values for each given y in the y space S2 The random variables X and Y are independent if and only if, or equivalently otherwise X and Y are said to be dependent Example 41-2: Let the joint Pmf of X and Y be defined by: = 1,2,3, y=1,2 Then and 2

3 Note that both f1(x) and f2(y) satisfy the properties of a pmf Since X and Y are dependent Example 41-3: Let the joint pmf of X and Y be The marginal pmfs are: and Then y=1,2; thus, X and Y are independent Example 41-4: Let the joint pmf of X and Y be f(x,y) = (1,1), (1,2), (2,2) Then the pmf of X is and that of Y is Thus, X & Y are dependent since for x=1,2 and y=1,2 Note: Let X1 and X2 be random variables of the discrete type with the joint pmf on the space S If u is a function of these two random variables, then if it exists, is called the mathematical expectation (or expected value) of Remark: must converge and be finite 3

4 Also, is a random variable - say with pmf g(y) on space S1, and it is true that Example 41-6: There are eight similar chips in a bowl: three marked (0,0), two marked (1,0), two marked (0,1), and one marked (1,1) A player selects a chip at random and is given the sum of the two coordinates in dollars If X1 and X2 represent those two coordinates, respectively, their joint pmf is Thus, The following mathematical expectations, if they exist, have special names: a) If is called the mean of Xi, i=1,2 b) If is called the variance of The mean can be computed from the joint pmf f or the marginal pmf The idea of joint distributions of two random variables of the discrete type can be extended to that of two random variables of the continuous type The definitions are really the same, except that integrals replace summations The joint pdf of two continuous-type random variables is an integrable function with the following properties: a) of X and Y b) 4

5 c) is an event defined in the plane Property (c) implies that P is the volume of the solid over the region A in the xy-plane and bounded by the surface The respective marginal pdf s of continuous-type random variables X&Y are given by: Where are the respective spaces of X & Y Example 41-8: Let X and Y have the joint pdf is the support; for example, P The marginal pdf s are given by: The expected values are: From these calculations, we see that E(X), E(Y), and E(Y 2 ) could be calculated from the marginal pdf s instead of the joint one Note: X and Y are independent if any only if the joint pdf factors into the product of their marginal pdf s; namely, 5

6 Thus X&Y in example 41-8 are dependent We now extend the binomial distribution to a trinomial distribution Here we have 3 mutually exclusive and exhaustive ways for an experiment to terminate: perfect, seconds, and defective In the n trials, let X1 = number of perfect items, X2=number of seconds, and X3=n-X1-X2= number of defectives Hence, the trinomial pmf is given by: where are non negative integers such that Example 41-10: In manufacturing a certain item, it is found that in normal production about 95% of the items are good ones, 4% are seconds, and 1% are defective A company has a program of quality control by statistical methods, and each hour an online inspector observes 20 items selected at random, counting the number X of seconds and the number Y of defectives Suppose that the production is normal Let us find the probability that, in this sample of size n=20, at least two seconds or at least two defective items are discovered If we let then =

7 42 The Correlation coefficient Definition: a If b If Remark: E Example 42-1: Let have the joint pmf The marginal pmfs are, respectively, Since of X1 are, respectively, are dependent The mean and the variance, and The mean and the variance of X2 are, respectively, 7

8 Cov Hence, Remark: The least squares regression line is: Example 42-2: Let X equal the number of ones and y the number of twos and threes when a pair of four-sided dice is rolled Then X and Y have a trinomial distribution with joint pmf where & are non negative integers Since the marginal pmf of X is 8

9 Also, Since, we have Cov(X,Y)= Using these values for the parameters, we obtain the line of best fit, namely, Remark: If X and Y are independent, then E(XY)=E(X)E(Y) ie But the converse is not necessarily true In other words, independence implies zero correlation, but zero correlation does not necessarily imply independence (Read example 42-3 in your text book) 43 Conditional Distributions Definition 43-1: The conditional probability mass function of X, given that Y=y, is defined by Similarly, the conditional probability mass function of Y, given that X=, is defined by Example 43-1: Let X & Y have the joint pmf In example 41-2, we showed that Thus, the conditional pmf of X, given that Y=y, is equal to when y =1 or 2 For example, 9

10 Similarly, the conditional pmf of Y, given that X=x, is equal to Moreover, Note that Thus, satisfies the conditions of a pmf Moreover, we can compute conditional probabilities such as, and conditional expectations such as Notes: 1 The conditional mean of Y, given that X=, is defined by: 2 The conditional variance of Y, given that X=, is defined by: which can be computed with Example 43-2: For example 43-1, compute when =3: Solution: and 10

11 Remark: In general, if E(Y/ ) is Linear, then If E(X/y) is linear, then Moreover, the product of the coefficient of in E(Y/ ) and the coefficient of y in E(X/ y) equals and the ratio of these two coefficients equals Definition: Let X and Y have a distribution of the continuous type with joint pdf f(, ) and marginal pdf s, respectively Then the conditional pdf, mean, and variance of Y, given that X=, are, respectively, a) b) and c) Example 43-5: Let X & Y be that rvs of example 41-8 Thus, We have h( / ) = Similarly, it can be shown that An illustration of a computation of a conditional probability is 11

12 Thus, the product of the coefficients of in, since each coefficient is positive Because the ratio of those coefficients is equal to 44 The Bivariate Normal Distribution Suppose we make the following three assumptions about the conditional distribution of Y, given that X= : a It is normal for each real b Its mean, E(Y/ ), is a linear function of c Its variance is constant; that is, it does not depend upon the given value of (a), (b), and (c) require that: Example 44-1: Let Assumptions (a), (b), and (c) imply that the conditional distribution of Y, given that X=, is The conditional mean line: =

13 Note: Suppose, we assume that the marginal pdf of X is: Hence, the joint pdf of X and Y is given by the product: Where A joint pdf of this form is called a bivariate normal pdf Example 44-2: Let us assume that in a certain population of college students, the GPAs say, X and Y in high school and the first year in college have an approximate bivariate normal distribution with parameters Since the conditional pdf of Y, given that X=32, is normal with mean, and standard deviation Theorem 44-1: If X and Y have a bivariate normal distribution with correlation coefficient then X and Y are independent if and only if Proof: If then the joint pdf factors into the product of the two marginal pdfs and hence X and Y are independent random variables Of course, if X 13

14 and Y are any independent random variables (not necessarily normal), then, if it exists, is always equal to zero 14