CHAPTER 5. Jointly Probability Mass Function for Two Discrete Distributed Random Variables:

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1 CHAPTER 5 Jointl Distributed Random Variable There are some situations that experiment contains more than one variable and researcher interested in to stud joint behavior of several variables at the same time. Jointl Probabilit Mass Function for Two Discrete Distributed Random Variables: Let X and Y are discrete random variables. The joint pmf p(x, ) is defined for each pair of numbers (x, ) b p(x, ) = P (X = x and Y = ), then the probabilit P [(X, Y ) A] can find b The marginal pmf of X and Y are P [(X, Y ) A] = (x,) p(x, ), A p X (x) = p(x, ) p Y () = x p(x, ) X and Y are independent, if for ever pair of x and p(x, ) = p X (x) p Y () Example The joint pmf of X and Y appears in the accompaning tabulation p(x,) x a. What is P (X = 1 and Y = 1)? b. Compute P (X 1andY 1). c. Give a word description of the event (X 0andY 0) and compute the probabilit of this event. d. Compute the marginal pmf of X and of Y. What is P (X 1)? e. Are X and Y independent r.v s? 1

2 Jointl Probabilit Densit Function for Two Continuous Distributed Random Variables: The joint pdf for two continuous random variables X and Y for an two-dimensional set A is P [(X, Y ) A] = f(x, )dxd If A be a rectangle {(x, ) : a x b, c d}, then P [(X, Y ) A] = P (a x b, c d) = The marginal pdf of X and Y are f X (x) = f Y () = f(x, )d f(x, )dx A b d for < x < for < < a c f(x, )ddx. Two continuous random variables X and Y are independent, if for ever pair of x and f(x, ) = f X (x)f Y () Example: Each front tire on a particular tpe of vehicle is supposed to be filled to a pressure of 26 psi. Suppose the actual air pressure in each tire is a random variable (X) for the right tire and (Y ) for the left tire, with joint pdf f(x, ) = { K(x ) 20 x 30, otherwise. a. What is the value of K? b. What is the probabilit that both tires are under filled? c. What is the probabilit that the difference in air pressure between the two tires is at most 2 psi? d. Determine the distribution of air pressure in the right tire alone. e. Are X and Y independent rv s? For two continuous rv s X and Y, the conditional pdf of Y given that X = x is If X and Y be discrete f Y X ( x) = p Y X ( x) = f(x, ) f X (x) p(x, ) p X (x) 2 < < < <

3 Expected Values, Covariance, and Correlation The expected value of function h(x, ) denoted b E[h(X, Y )] or µ h(x,y ) is { x E[h(X, Y )] = h(x, )p(x, ) h(x, )f(x, )dxd The covariance between two random variables X and Y is if X and Y are discrete if X and Y are continuous Cov(X, Y ) = E[(X µ X )(Y µ Y )] { x = (x µ X)( µ Y )p(x, ) (x µ X)( µ Y )f(x, )dxd X and Y discrete X and Y continuous Also Cov(X, Y ) = E(XY ) µ X µ Y The correlation coefficient of two random variables is and has the following properties Corr(X, Y ) = ρ X,Y = Cov(X, Y ) σ X σ Y Corr(aX + b, cy + d) = Corr(X, Y ), if a and c have same sign (same positive or negative). 1 ρ X,Y 1 ρ X,Y = 1 or -1 if and onl if Y = ax + b such that a 0 If X and Y are independent ρ = 0 - Example: Consider the following joint pmf p(x,) x a. What is E(X + Y )? b. What is expected value for maximum of X and Y? c. Compute the covariance for X and Y. d. Compute ρ for X and Y. 3

4 The Distribution of the Sample Mean A statistic is an quantit that calculated from sample like sample mean ( X). Random variables X 1, X 2, X n from a random sample of size n if 1. The X i s are independent random variables. 2. Ever X i has the same probabilit distribution. If X 1, X 2, X n be a random sample from a distribution with mean µ and variance σ 2, then 1. E( X) = µ X = µ X is unbiased 2. V ( X) = σ 2 x = σ2 n Also, for T = X 1 + X X n (the total sample) 1. E(T ) = nµ 2. V (T ) = nσ 2 If X 1, X 2, X n be a random sample from a normal distribution with µ and σ 2, then for an n, sample mean is normall distributed with µ and σ 2, i.e., also X N(µ, σ2 n ) T N(nµ, nσ 2 ) The Central limit theorem For a random sample X 1, X 2, X n from a distribution with µ and σ 2, sample mean has approximatel a normal distribution with mean µ and variance σ2, if n is sufficientl large. n (Also total sample has a normal distribution) If n 30, the central limit theorem can be used. - Example: The inside diameter of a randoml selected position ring is a random variable with mean value 12 cm and standard deviation 0.04 cm. a. If X is the sample mean for a random sample of n = 16 rings, where is the sampling distribution of X centered, and what is the standard deviation of the X distribution? b. Answer the question part (a) for a sample size of n = 64 rings. c. For which of the two random samples, X is more likel to be within 0.01 cm of 12 cm? d. Calculate P (11.99 X 12.01) when n = 64. 4

5 The Distribution of a Linear Combination In general a 1 X 1 + a 2 X a n X n is a linear combination of random variables X 1, X 2,, X n have mean values µ 1, µ 2,, µ n, and variance of σ 2 1, σ 2 2,, σ 2 n. respectivel E(a 1 X 1 + a 2 X a n X n ) = a 1 E(X 1 ) + a 2 E(X 2 ) + + a n E(X n ) = a 1 µ 1 + a 2 µ a n µ n, n n V (a 1 X 1 + a 2 X a n X n ) = a i a j Cov(X i X j ). i=1 If X i s and X j s be independent, Cov(X i, X j ) = 0, then V (a 1 X 1 + a 2 X a n X n ) =? In particular, for difference of two random variables j=1 E(X 1 X 2 ) = E(X 1 ) E(X 2 ) V (X 1 X 2 ) = V (X 1 ) + V (X 2 ), if X 1 and X 2 are independent If X 1, X 2,, X n are independent and normall distributed, an linear combination of them has also normal distribution. Example: Let X 1, X 2, X 3, X 4, X 5 be the observed numbers of miles per gallon for the five cars. suppose these variables are independent and normall distributed with µ 1 = µ 2 = 20, µ 3 = µ 4 = µ 5 = 21, and σ 2 = 4 for X 1 and X 2 and σ 2 = 3.5 for others, define Y as Y = X 1 + X 2 2 X 3 + X 4 + X 5 3 Compute P (0 Y ) and P ( 1 Y 1). Suggested Exercises for Chapter 5: 3, 5, 11, 13, 15, 19, 25, 27, 31, 37, 39, 41, 47, 49, 51, 55, 59, 63, 65, 69, 73, 75, 5