Probability and Stats question...please help!?

Question

I need to know how to derive the Moment Generating Functions of the normal and the chi-squared distribution. If anyone does know how to do this, please help. I get stuck when i do it myself. I would appreciate any insight/help!

Thank You.

Answer 1

The normal density is continuous=f(x)

= (1/(sigma*sqrt(2*pi) ) *exp[-(1/sigma^2)*(x-mu)^2]

the moment generating function is

m(t) = E(e^tX) =  integral -inf to inf  of (e^tx * f(x) )dx

No exotic integration is necessary.  You will have something like x^2+x  in the integrand.   Just complete the square and rearrange until you see that what is left inside the integral is just a normal denity function integrated from -ing to inf.  That must be =1.  So you are left with something else that you factored out.

m(t) = exp (mu*t + t^2*sigma^2/2)



Look up the gamma density.  This is easier since  x will be to the first power.   There are two parameters alpha and beta.  When alpha=df/2 and beta=2 this is the chisq.  You will again notice that you integrate a density completely and it becomes = 1.  

m(t) = (1 - 2t)^ (-df/2)   where   df  is degrees of freedom.