STAT 350: Lecture 7
In class I started discussion of the Normal distribution. I computed the
mean and variance of a standard normal and then of 
.
Here I will just show you a few more integrals:
The kth moment of a standard normal is
We can do this integral by parts with 
and
. This makes 
and
so that by integration by parts
Since 
and 
we see that
if k is odd and 
if k is even.
We can also compute the moment generating function of Z, that is,
by writing
and substituting u=z-t, du=dz in the integral to see that
Now if 
then 
so
that the 
central moment of X, namely 
is 0 for odd k and 
for k even.
Similarly
is the moment generating function of X.