Would you please explain all steps involved in Moment generating function of normal distribution especially how Sqrt2Pi getting cancel?

Answer 1

I could not find a good derivation of the mgf for the normal on the web. However, I found a complete derivation on page 270 of Probability and Statistics by DeGroot, 2002 Edition. Most university libraries should have a copy.

The first equation on page 270 goes from exp(a)exp(b) to exp(a+b).

The second equation is a direct result of "completing the squares" method. See page 281.

In third equation on page 270 DeGroot goes from exp(a+b) to exp(a)exp(b) which allows to define a constant exp(mu*t +1/2*sigma^2 * t^2) which he can pull out from the integral.

The "disappearing" of the square-root with 2pi occurs because the C as defined in DeGroot iin the fourth equation is the complete integral of the cdf of a normal distribution, for any value of t.

DeGroot states we can replace mu with mu+sigma^2*t, but I would have preferred

Let mu* = mu +sigma^2*t, then for any t, the cdf of the normal, N(mu*, sigma), integrated from minus infinity to positive infinity must equal 1.

Thus what appeared to be an integration requires no integration at all. I could restate the derivation as:

mgf(t) = f(t) * integral of N(x, mu*,sigma) from minus infinity to positive.

mgf(t) = f(t) * 1

mgf(t) = exp(mu*t +1/2*sigma^2 * t^2)