moment generating function

Question

Let f(x)=4 e^(-4x) if 0<x<infinty, and zero otherwse:

 

 Use the moment generating function to find E(x) and V(x)?

 

My solution:

E(x)=m'x(t) when t=0

so E(x)=1/4

E(x^2)=m''x(t) when t=0

E(x^2)=-2/16

v(x)=E(x^2)-E(x)=(-2/16)-(1/4)

 

but my teacher told me it was wrong answer

Could you help me ?

Answer 1

The moment generating function is

 

M(t) = E(e^tx) = Integral form -∞ to ∞ of e^txf(x)dx.  We can find different moments by taking the derivative of the moment generating function.

 

E(x^r) = d^rM(t)/dt^r

 

The first derivative is the mean or expected value Ex).  The second derivative is E(x²) and V(x) = E(x²) - u²

 

First find the moment generating function.

 

M(t) = 4*Integral from 0 to ∞ of e^tx*e^-4xdx = 4*Integral from 0 to ∞ of e^-(4-t)xdx = 4/(4-t)

 

Taking the first derivative of M(t) gives

 

E(x) = 4/(4-t)² @ t = 0 → E(x) = 1/4

 

Taking the second derivative gives

 

E(x²) = 8/(4-t)3 @ t = 0 → E(x²) = 1/8

 

V(x) = 1/8 - (1/4)² = 1/16