Question

s Let X be a discrete random variable with probability mass function: f(x) = (1/3)*(3/4)^x for x = 1, 2, 3, . . .. (a) It can be shown that the moment generating function (MGF) exists for X if t < − ln(3/4). Derive the moment generating function, M(t). Note: Even if you recognize this distribution, you are graded on how you derive the MGF mathematically.

(b)using the mgf derive the mean and the variance of X

Answer 1

here mgf Mx(t)= = = =(1/3)*(3e^t/4)/(1-3e^t/4)

Mx(t)=e^t/(4-3e^t)

b)

first derivative of mgf :M'_x(t)=(d/dt)*e^t*(4-3e^t)^-1= e^t*(4-3e^t)^-1 +3e^2t*(4-3e^t)^-2

hence mean =E(X)=M'_x(0)=e⁰*(4-3e⁰)^-1 +3e^2*0*(4-3e⁰)^-2 =1+3 =4

second derivative of mgf :M''_x(t)=(d/dt)*e^t*(4-3e^t)^-1 +3e^2t*(4-3e^t)^-2 =e^t*(4-3e^t)^-1 +3e^2t*(4-3e^t)^-2 +6e^2t*(4-3e^t)^-2+18e^3t*(4-3e^t)^-3

hence E(X²)=M''_x(0) =e⁰*(4-3e⁰)^-1 +3e^2*0*(4-3e⁰)^-2 +6e^2*0*(4-3e⁰)^-2+9e^3*0*(4-3e⁰)^-3 =1+3+6+18=28

hence Variance =E(X²)-(E(X))² =28-4² =28-16 =12