STAT 400 Lecture AL1 Examples for 4.1, 4.4 (Part 2) Spring 2015 Dalpiaz Independent Random Variables 1. Consider the following joint probability distribution p ( x, y ) of two random variables X and Y: x \ y 0 1 2 1 0.15 0.10 0 0.25 2 0.25 0.30 0.20 0.75 0.40 0.40 0.20 Recall: A and B are independent if and only if P ( A B ) = P ( A ) P ( B ). a) Are events {X = 1} and {Y = 1} independent? Def Random variables X and Y are independent if and only if discrete p ( x, y ) = p X ( x ) p Y ( y ) for all x, y. continuous f ( x, y ) = f X ( x ) f Y ( y ) for all x, y. F ( x, y ) = P ( X x, Y y ). f ( x, y ) = 2 F ( x, y )/ x y . Def Random variables X and Y are independent if and only if F ( x, y ) = F X ( x ) F Y ( y ) for all x, y. b) Are random variables X and Y independent?2. Let the joint probability density function for ( X , Y ) be otherwise01 ,10 ,10 60 ,2 yxyxyxyxf Recall: f X ( x ) = 2 2 1 30 xx , 0 < x < 1, f Y ( y ) = 3 1 20 yy , 0 < y < 1. Are random variables X and Y independent? 3. Let the joint probability density function for ( X , Y ) be otherwise010 ,10 ,yxyxyxf Are X and Y independent? 4. Let the joint probability density function for ( X , Y ) be otherwise00 ,101 12 ,2 yxxxyxfye Are X and Y independent?If random variables X and Y are independent, then E ( g ( X ) h ( Y ) ) = E ( g ( X ) ) E ( h ( Y ) ). 5. Suppose the probability density functions of T 1 and T 2 are f T 1 ( x ) = e – x, x > 0, f T 2 ( y ) = e – y, y > 0, respectively. Suppose T 1 and T 2 are independent. Find P ( 2 T 1 > T 2 ). 6. Let X and Y be two independent random variables, X has a Geometric distribution with the probability of “success” p = 1/3 , Y has a Poisson distribution with mean 3. That is, p X ( x ) = 1 3231x, x = 1, 2, 3, … , p Y ( y ) = ! 3 3yey, y = 0, 1, 2, 3, … . a) Find P ( X = Y ). b) Find P ( X = 2 Y
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