STAT 400 Lecture AL1 Spring 2015 Dalpiaz Examples for 4 1 4 4 Part 2 Independent Random Variables 1 Consider the following joint probability distribution p x y of two random variables X and Y x Recall a Def 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 A and B are independent if and only if P A B P A P B Are events X 1 and Y 1 independent Random variables X and Y are independent if and only if discrete p x y p X x p Y y continuous f x y f X x f Y y F x y P X x Y y Def for all x y 2 f x y F x y x y Random variables X and Y are independent if and only if F x y F X x F Y y b for all x y Are random variables X and Y independent for all x y 2 Let the joint probability density function for X Y be 2 f x y 60 x y 0 x 1 0 y 1 x y 1 Recall 0 otherwise f X x 30 x 2 1 x 2 0 x 1 f Y y 20 y 1 y 3 0 y 1 Are random variables X and Y independent 3 Let the joint probability density function for X Y be x y 0 x 1 0 y 1 otherwise 0 f x y Are X and Y independent 4 Let the joint probability density function for X Y be f x y 12 x 1 x e Are X and Y independent 0 2 y 0 x 1 y 0 otherwise 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 1 2 p X x 3 3 pY y 3 y e 3 a Find P X Y b Find P X 2 Y y x 1 x 1 2 3 y 0 1 2 3
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