2 The moment generating function Mxt for a random variable X

2. The moment generating function Mx(t) for a random variable X is Mx(t)= e^{4e^t-4}

(a) Identify the distribution and the parameter(s) of the distribution.

(b) Determine the mean and variance of X.

(c) Determine Pr ( X > 3 ).

Solution

mgf= e^ ( 4*e^(t)-4 ) = e^ ( 4*(e^t - 1) )
it\'s a poisson distributin with = 4....

b) mean = 4 and variance =4......

c) Pr[ X >3] =1 - P [ X <=3 ] = 1 - P [ X=0 ] - P[X=1] -P [X=2] - P[X=3].....
= 1 - e^(-4)*4^0 / 0! - e^(-4)* 4 / 1! - e^(-4) * 4^2 /2! - e^(-4)*4^3 / 3! = 0.43347.....