概率论习题

1.The joint density of X and Y is given by：

f(x,y)?C(x?y)e?y?y?x?y,0?y??

(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：

?x??cy0?x?1,1?y?5f(x,y)??5

?otherwise?0where c is a constant.

(a) What is the value of c? (b) Are X and Y independent? (c) Find P{X?Y?3}.

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 U1 and U2be independent and uniform on [0, 1]. Find and sketch the density function of S?U1?U2.

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

??z1fZ(z)??f(y,)dy

??y|y|6.If X and Y are independent standard normal random variables, find P{X2?Y2?1}. 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.