Let X have a uniform distribution on the interval (0, 1). Given X = x, let Y have

a uniform distribution on (0, x). (a)TheconditionalpdfofY,giventhatX=x,isfY|X(y|x)=1 for0

x

since Y|X ? U(0,X). Show that the mean of this (conditional) distribution is E(Y |X) = X , and hence, show that EX{E(Y |X)} = 1. [Hint: what is the

mean of X ?] 2

1. (b)Noting that fY |X (y|x) = f (x, y)/fX (x), determine the joint pdf of X and Y .
2. (c)ShowthatthemarginalpdfofY isfY(y)=?lny. [Hint:
   

5. Let X have a uniform distribution on the interval (0,1). Given X = 3:, let Y have a uniform distribution on (0, :6). (a) The conditional pdf of Y, given that X = :13, is fy|X(y|x) = i for 0