Normal Random Variables
Lecture XII
I. Univariate Normal Distribution.
A. Definition 5.2.1. The normal density is given by
2
2
11
exp , 0
2
2
x
f x x
When
X
has the above density, we write symbolically
2
~,XN
.
B. Theorem 5.2.1. Let
X
be
2
,N
. The
EX
and
2
VX
.
2
2
2
11
exp 2
2
x
E X x dx
1. Using the change in variables technique, we create a new random variable
z
such that
x
z x z
dx dz
Substituting into the original integral yields:
2
22
2
11
exp 2
2
11
exp 2
2
11
exp 2
2
E X z z dz
z z dz
z dz
Taking the integral of the first term first, we have:
2 2 2
2
1 1 1
exp exp
22
2
1
exp 0
2
z z dz C z z dz
Cz
