STAT 400 1 Spring 2015 Discussion 8 Suppose that the random variables X and Y have joint p d f f x y given by f x y C x 2 y 0 x y x y 2 zero elsewhere a Sketch the support of X Y That is sketch 0 x y x y 2 b What must the value of C be so that f x y is a valid joint p d f c Find P Y 2 X d Find P X Y 1 e Find the marginal probability density function for X f Find the marginal probability density function for Y Hint Consider two cases 0 y 1 and 1 y 2 g Find E X h Find E Y i Find E X Y 2 Let X and Y have the joint probability density function f x y C 0 x 1 0 y x 1 x zero elsewhere That is suppose that X Y is uniformly distributed over the region defined by 0 x 1 0 y x 1 x a Find the value of C so that f x y is a valid joint p d f 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 Are X and Y independent If not find Cov X Y 3 Every month the government of Neverland spends X million dollars purchasing guns and Y million dollars purchasing butter Assume X and Y are independent X has a Normal distribution with mean 265 and standard deviation 40 in millions of dollars and Y has a Normal distribution with mean 170 and standard deviation 30 in millions of dollars a Find the probability that the government of Neverland spends more on guns than on butter during a given month That is find P X Y b Find the probability that the government of Neverland spends more on guns than twice the amount it spends on butter during a given month That is find P X 2 Y c Find the probability that the government of Neverland exceeds the 500 million spending limit during a given month That is find P X Y 500 1 Suppose that the random variables X and Y have joint p d f f x y given by f x y C x 2 y 0 x y x y 2 Sketch the support of X Y a That is sketch 0 x y x y 2 What must the value of C be b so that f x y is a valid joint p d f Must have f x y dx dy 1 1 2 x C x 2 y dy dx 0 x 1 C 2 2 y 2 x x y dx y x 2 0 1 C 2 x 2 0 2 C x 1 2 x 2 2 x 2 dx 2 C x 3 dx 0 C 2C 3 C 4 1 1 x x 6 2 3 0 C 6 zero elsewhere Find P Y 2 X c x y 2 2 4 y 3 3 x y 2x 2 3 2 x 2 1 6 x y dy dx 0 2x 2 3 1 0 2 3 1 3 x 2 y 2 yy 22 xx dx 3 x 2 2 x 2 4 x 2 dx 0 2 3 1 12 x 2 12 x 3 9 x 4 dx 0 3 4 9 2 9 2 2 2 3 1 4x3 3x4 x5 1 4 3 5 3 5 3 3 0 OR 1 2 x 2 2 6 6 x x y y dy dy dx dx 2 3 x 0 x 2 3 2 x OR 4 3 2 y y 2 2 6 x y dx dy 6 x y dx dy 1 y 2 0 y 2 1 OR 2 2 y y 2 2 2 6 x y dx dy 6 x y dx dy 0 0 4 3 0 4 3 1 5 87 135 d Find P X Y 1 0 5 1 x 6 x 2 y dy dx 0 x 0 5 0 3 x 2 y 2 y y 1 x x dx 2 2 2 3 x 1 x x dx 0 5 0 3 x 0 5 2 6 x 3 dx 0 3 0 5 x3 x4 2 0 3 3 1 1 2 2 2 e 4 1 3 1 0 03125 8 32 32 Find the marginal probability density function for X First X can only take values in 0 1 f X x f x y dy 3x2 2 x 2 6 x y dy x 3 x 2 y 2 y y 2 x x 2 x 2 x 2 12 x 2 12 x 3 12 x 2 1 x 0 x 1 f Find the marginal probability density function for Y Hint Consider two cases 0 y 1 and 1 y 2 First Y can only take values in 0 2 y 6 x 2 y dx 0 f Y y f x y dx 2 y 6 x 2 y dx 0 0 y 1 1 y 2 x y 2 x3 y x 0 x 2 y 3 2x y x 0 2y4 2 y 2 y 3 g x f X x dx 0 y 1 1 y 2 1 x 12 x 2 1 x dx 0 60 0 Find E Y y f Y y dy E Y i 1 y 2 Find E X E X h 0 y 1 1 0 y 2 y dy 4 2 y 2 y 2 y 3 dx 1 Find E X Y 1 2 x E X Y 22 2 x y 6 x y dy dx 35 0 x 1 11 16 15 3 15 2 Let X and Y have the joint probability density function 0 x 1 0 y x 1 x f x y C zero elsewhere That is suppose that X Y is uniformly distributed over the region defined by 0 x 1 0 y x 1 x a Find the value of C so that f x y is a valid joint p d f 1 x 1 x C dy dx 0 Must have 1 0 1 C x x 2 dx 0 b C 6 C 6 Find the marginal probability density function of X f X x x 1 x f X x 6 dy 6 x 1 x 0 0 x 1 c Find the marginal probability density function of Y f Y y y x 1 x x 1 x x 2 where x1 x2 fY y 6 dx x1 6x x2 x1 12 1 2 1 y 4 1 y 6 1 4y 4 x2 1 2 0 y 1 4 1 y 4 d Are X and Y independent If not find Cov X Y f x y f X x f Y y X and Y are NOT independent OR The support of X Y is NOT a …
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