STAT 400 Lecture AL1 1 Spring 2015 Dalpiaz Practice Problems 4 Let X and Y have the joint p d f f X Y x y C x 2 y 3 0 x 1 0 y x zero elsewhere a What must the value of C be so that f X Y x y is a valid joint p d f b Find P X Y 1 c Let 0 a 1 Find P Y a X d Let a 1 Find P Y a X e Let 0 a 1 Find P X Y a f Find f X x g Find E X h Find f Y y i Find E Y j Find E X Y k Find Cov X Y l Are X and Y independent 2 Let X and Y have the joint probability density function f X Y x y 1 x x 1 0 y 1 x zero elsewhere a Find f X x b Find E X c Find f Y y d Find E Y 3 Let X and Y have the joint probability density function f X Y x y 1 x 0 x 1 0 y x zero elsewhere a Find f X x b Find E X c Find f Y y d Find E Y e Find P X Y 1 f Find Cov X Y 4 Let X and Y be two random variables with joint p d f f x y 64 x exp 4 y 64 x e 4 y 0 x y zero elsewhere a Find P X 2 Y b Find the marginal p d f f X x of X c Find the marginal p d f f Y y of Y d Are X and Y independent If not find Cov X Y and Corr X Y e Let a 1 Find P Y a X f Let a 0 Find P X Y a 5 Let the joint probability mass function of X and Y be defined by p x y x y 32 x 1 2 a Find P Y X b Find p X x the marginal p m f of X c Find p Y y the marginal p m f of Y d Are X and Y independent If not find Cov X Y 6 y 1 2 3 4 Let the joint probability mass function of X and Y be defined by p x y x y 30 x 1 2 a Find P Y X b Find p X x the marginal p m f of X c Find p Y y the marginal p m f of Y d Are X and Y independent If not find Cov X Y y 1 2 3 4 7 Suppose the joint probability density function of X Y is f x y C x y 2 0 0 y x 1 otherwise a Find the value of C that would make f x y a valid probability density function b Find the marginal probability density function of X f X x c Find the marginal probability density function of Y f Y y d Find P X 2 Y f Are X and Y independent If not find Cov X Y 8 Let X and Y have the joint probability density function f x y C x e Find P X Y 1 0 x 1 0 y x 1 x zero elsewhere a Find the value of C so that f x y is a valid joint p d f b Find f X x c Find E X d Find f Y y e Find E Y f Are X and Y independent 9 Suppose that X Y is uniformly distributed over the region defined by x 0 y 0 x 2 y 2 1 That is f x y C a x 0 y 0 x 2 y 2 1 zero elsewhere What is the joint probability density function of X and Y That is find the value of C so that f x y is a valid joint p d f b Find P Y 2 X d Are X and Y independent c Find P X Y 1 1 Let X and Y have the joint p d f f X Y x y C x 2 y 3 a 0 x 1 0 y x zero elsewhere What must the value of C be so that f X Y x y is a valid joint p d f 1 C 4 dx 4 x dx 0 1 x 2 3 C x y dy 0 0 b C 1 20 C 20 Find P X Y 1 y x and y 1 x x y 2 and x 1 y 5 1 2 y 5 1 2 P X Y 1 0 5 1 2 0 5 1 2 0 1 y 2 3 20 x y dx dy y2 20 1 y 3 y 3 20 y 9 dy 3 3 20 20 20 3 y 20 y 4 20 y 5 y 6 y 9 dy 3 3 3 10 6 20 7 2 10 5 y4 4y5 y y y 3 21 3 3 5 1 2 0 030022 0 OR 2 y 5 1 1 x 2 x and y 1 x 5 1 3 5 2 2 x 2 3 P X Y 1 1 20 x y dy dx 3 5 1 x 1 2 5 x 1 1 4 5 x 2 1 x 4 dx 3 5 2 5 x 1 1 2 20 x 3 25 x 4 20 x 5 5 x 6 dy 3 5 2 1 10 5 5 1 x 3 5 x 4 5 x 5 x 6 x 7 0 030022 7 3 3 5 3 2 c Let 0 a 1 Find P Y a X ax 2 3 P Y a X 20 x y dy dx 0 0 1 1 5a 0 4 x 6 dx 5 4 a 7 d Let a 1 Find P Y a X y x and y a x x a y a 1 a 2 3 P Y a X 1 20 x y dx dy 1 0 y2 0 1 1 a2 y 6 20 y 20 y 9 3a 3 3 1 a dy 1 2 7 a 10 1 a2 a2 x 2 5 x 4 5 a 4 x 6 dx 1 P Y a X 1 20 x 2 y 3 dy dx 1 10 a 7 0 0 a x 1 e Let 0 a 1 Find P X Y a 1 P X Y a 1 a23 y x and y x a 2 3 x 1 2 3 20 x y dy dx 1 a x a23 a x 4 5 x 4 5 a x2 dx 1 a 4 1 x 5 5 6 a 10 3 5 a 4 x a23 f Find f X x x 20 x f X x 2 y 3 dy 5 x 4 0 x 1 0 g …
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