Question

1. This question is on probability a. Suppose that X is a normally distributed random variable, where X N (M, o). Show that E [cºX f (x)] = cºu+20oʻE [ f (x + 002)] where f is a suitable function and 0 € R is a scalar. Hint: Write X = 1 +o0; 0~ N (0,1) and calculate the resulting integral b. Consider the probability density function X>0 p(x) = { Az exp (-1.2-2) 10 x < 0 (>0) is a parameter and 2 is a variable. You are given where that A=4 / 4 If E. denotes the expectation, show that E[X] = By using integration by parts, or otherwise, deduce that for n = 0,1,2, ... the even moments of this distribution are given by 1.3... (2n +1) E [X2n] =? (2)" and the odd moments are given by 2 (n + 1)! E [X2+] = T (2n+1)/2

Answer 1

a) The PDF of is

The expectation,

Let .

The integral becomes,

Let .

Now let

The proof is complete.

b) The PDF is .

The expectation is

Let .

The proof is complete.

The even moments are

Let . Use integration by parts.

The proof is complete.

The odd moments are (Use Gamma function)

Let .

All the proofs are complete.