1. Let a random variable X has the CDF as;

FX(x) = {

1 ; x 1

1

2

+

x

2

; 0 x < 1

0 ; x < 0.

Verify that FX(x) is a CDF. Also- (i) find the PDF of X, (ii) Calculate E[e

X

], (iii) Determine

the probability P[X = 0|X 0.5].

2. Let us consider a discrete random variable having the pmf given by,

PX(k) =

{

1

3

, k = 1

2

3

, k = 2.

0, k = 3

Obtain the moment generating function for X. Also get the values of its mean and

variance.