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1.The joint density of X and Y is given by ：
∞<<<<--=-y y x y e y x C y x f y 0,)(),(
(a) Find C .
(b) Find the density function of X .
(c) Find the density function of Y .
2. Let X and Y be continuous random variables with joint density function ：
⎪⎩⎪⎨⎧<<<<+=otherwise y x cy x y x f 0
51,105),(
where c is a constant.
(a) What is the value of c ?
(b) Are X and Y independent?
(c) Find }3{>+Y X P .
3.Let X and Y be independent uniform (0, 1) random variables.
(a) Find the joint density of U = X ,V = X + Y .
(b) Use the result obtained in part (a) to compute the density function of V .
4.Let 1U and 2U be independent and uniform on [0, 1]. Find and sketch the density function of 21U U S +=.
5.Let X and Y have the joint density function f (x, y), and let Z = XY . Show that the density function of Z is
y y y z y f z f Z d |
|1),()(⎰+∞∞-= 6.If X and Y are independent standard normal random variables, find }1{22≤+Y X P .
7.Find the joint density of X + Y and X/Y , where X and Y are independent exponential random variables with parameter λ. Show that X + Y and X/Y are independent.