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1. (5pts) Suppose X and Y are random variables with joint probability density function 2vxe x+y 3 f F(x,y) = , X > 0,y > 0 3VII 0, otherwise. Are X and Y independent? Explain. 2. Suppose X and Y have the joint probability function f(x,y)=(cl2x+ = y), 0, + 0 < X < 1,0 < y < 2 otherwise. a. (5pts) Find the value of C. b. (8pts) Find the marginal distribution of Y. c. (10pts) Find the joint distribution function F(x,y). 3. Suppose X and Y have joint probability density function 3y f F(x,y) = , < y < X < 2, , 4 0, otherwise. a. (10pts) Find P(Y < 1). b. (10pts) Find E(Y|X = x). c. (10pts) Find P(Y < 0.5|X=1). 4. Assume that X ~ Exp(1) and Y ~ Exp( (1) are two independent variables. Let S = Y and T = X + XY. Find a. (5pts) E (E(T|S)) b. (10pts) V(T) 5. Let Y1, Y2, , Yn be a random sample from a distribution with the probability density function y > 1 = otherwise, where 0 > 0. a. (7pts) Find the method of moments estimator of 0. b. (10pts) Find the maximum likelihood estimator for 0. 6. (10pts) Let Y1, , Y2, ,Yn , be a random sample from the uniform distribution on the interval (0, 10 - 20) where 10 < 0 < 5. Let 0 = 10 - 2Y. Show that 0 is consistent for 0. 10

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