5.5Normal Random Variables 169

The moment generating function of a normal random variable with parametersμand
σ^2 is derived as follows:

φ(t)=E[etX]

=

1

√

2 πσ

∫∞

−∞

etxe−(x−μ)

(^2) /2σ 2
dx

1
√
2 π
eμt
∫∞
−∞
etσye−y
(^2) /2
dy by lettingy=
x−μ
σ

eμt
√
2 π
∫∞
−∞
exp
{
−
[
y^2 − 2 tσy
2
]}
dy

eμt
√
2 π
∫∞
−∞
exp
{
−
(y−tσ)^2
2

• t^2 σ^2
  2
  }
  dy
  =exp
  {
  μt+
  σ^2 t^2
  2
  }
  1
  √
  2 π
  ∫∞
  −∞
  e−(y−tσ)
  (^2) /2
  dy
  =exp
  {
  μt+
  σ^2 t^2
  2
  }
  (5.5.1)
  where the last equality follows since
  1
  √
  2 π
  e−(y−tσ)
  (^2) /2
  is the density of a normal random variable (having parameterstσand 1) and its integral
  must thus equal 1.
  Upon differentiating Equation 5.5.1, we obtain
  φ′(t)=(μ+tσ^2 ) exp
  {
  μt+σ^2
  t^2
  2
  }
  φ′′(t)=σ^2 exp
  {
  μt+σ^2
  t^2
  2
  }
  +exp
  {
  μt+σ^2
  t^2
  2
  }
  (μ+tσ^2 )^2
  Hence,
  E[X]=φ′(0)=μ
  E[X^2 ]=φ′′(0)=σ^2 +μ^2
  and so
  E[X]=μ
  Var(X)=E[X^2 ]−(E[X])^2 =σ^2
  Thusμandσ^2 represent respectively the mean and variance of the distribution.