Distribution functions (Rayleigh distribution)

[Meital]

I am trying to solve some problems on distribution functions and expected values, and I am stuck with 4 of these problems. I will appreciate any help or hints. thanks in advance!

 

Suppose X_1 and X_2 are independent and normal(0,1). Find the distribution function of Y = ( X_1^2 + X_2^2)^1/2. This is the Rayleigh distribution.

 

Suppose X1...Xn are independent and have density function f(x). Let Y =min { X1,...,Xn} and Z= max{X1,...,Xn}. Compute P( Y >= y, Z=<z) and differentiate to find joint density of Y and Z.

 

Suppose X has the standard normal distribution. Compute E|X|.

 

Suppose X has the standard normal distribution. Use integration by parts to conclude that EX^k = (k-1)EX^(k-2) and then conclude that for all integers n, EX^(2n-1)=0 and EX^2n = (2n-1)(2n-3)...3.1.