STAT 400 Lecture AL1 Spring 2015 Dalpiaz Answers for 4 1 4 4 2 Independent Random Variables 1 Consider the following joint probability distribution p x y of two random variables X and Y x Recall a 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 P X 1 Y 1 p 1 1 0 10 0 25 0 40 P X 1 P Y 1 X 1 and Y 1 are independent Def 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 for all x y Are random variables X and Y independent p 1 0 0 15 0 25 0 40 p X 1 p Y 0 X and Y are NOT independent 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 The support of X Y is not a rectangle X and Y are NOT 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 X and Y are NOT independent 4 Let the joint probability density function for X Y be f x y 12 x 1 x e 2 y 0 0 x 1 y 0 otherwise Are X and Y independent f X x 12 x 1 x e 2 y dy 6 x 1 x 0 x 1 0 1 f Y y 12 x 1 x e 2 y dx 2 e 2 y y 0 0 Since f x y f X x f Y y for all x y X and Y are 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 x y y x dx dy P 2 T 1 T 2 e dx dy e e 0 y 2 0 y 2 e 0 y e 2 2 y dy dy e 2 y 2 0 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 x 1 1 2 p X x pY y a 3 y e 3 y 0 1 2 3 y x 1 2 3 3 3 Find P X Y 1 2 k 1 3 3 p X k p Y k P X Y k 1 e 3 2 k 1 k 1 e 1 e 3 b 2 k k 1 2k 1 2 k 0 k e 3 3 k e 3 k e 3 2 e2 1 0 159 Find P X 2 Y P X 2 Y k 1 p X 2 k p Y k e 3 4 k 1 k 1 3 2 k e 3 2 1 2 k 1 3 3 e4 3 1 2 k 1 3 k e 3 0 069544 For fun c P X Y 1 2 3 3 y 0 x y 1 2y e 3 y 0 y e 1 x 1 3 y e 3 y k 2 3 y 0 y 3 y e 3 y
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