Question

The Moment Generating function of X is given by
M (t) X = 2
(1 )
1
 t
, Find E(X2
)

Answer 1

Answer:

If X∼N(0,1), integrate to find the moment generating function of a random variable X2 and identify the distribution of X2 using the moment generating function.

E[etX2]=∫∞−∞etx2e−x22π−−√dx

which reduces to

=12π−−√∫∞−∞etx2e−x2dx=12π−−√∫∞−∞ex2(t−12)dx

and thus I am stuck. I'm sure there must be some trick to this (like completely the square for the mgf of a standard normal variable X) but I can't figure out what it might be.