Question

6. (4 marks) The moment generating function (mgf) of a random variable X is given by m(t)-e2 (a) Use the mgf to find the mean and variance of X (b) What is the probability that X-2?

Answer 1

a)

here first derivative of mgf m'(t)=(d/dt)*m(t) =(d/dt)*e^2t =2e^2t

therefore mean E(X)=m'(0)=2e^2*0 =2

second derivative of mgf m''(t)=(d/dt)*m'(t)=(d/dt)*2e^2t =4e^2t

hence E(X²) =m''(0)=4e^2*0 =4

hence Variance =E(X²)-(E(X))² =4-2² =0

b)

as mgf =p_x1*e^tx1+p_x2*e^tx2+p_x3*e^tx3+...

comaring it with given mgf:

P(X=2)=1