170 Chapter 5: Special Random Variables

An important fact about normal random variables is that ifXis normal with meanμ
and varianceσ^2 , thenY =αX+βis normal with meanαμ+βand varianceα^2 σ^2.
That this is so can easily be seen by using moment generating functions as follows.

E[et(αX+β)]=etβE[eαtX]

=etβexp{μαt+σ^2 (αt)^2 /2} from Equation 5.5.1

=exp{(β+μα)t+α^2 σ^2 t^2 /2}

Because the final equation is the moment generating function of the normal random
variable with meanβ+μαand varianceα^2 σ^2 , the result follows.
It follows from the foregoing that ifX∼N(μ,σ^2 ), then

Z=

X−μ

σ

is a normal random variable with mean 0 and variance 1. Such a random variableZis
said to have astandard,orunit, normal distribution. Let (·) denote its distribution
function. That is,

(x)=

1

√

2 π

∫x

−∞

e−y

(^2) /2
dy, −∞<x<∞
This result thatZ=(X−μ)/σhas a standard normal distribution whenXis normal
with parametersμandσ^2 is quite important, for it enables us to write all probability
statements aboutXin terms of probabilities forZ. For instance, to obtainP{X<b},we
note thatXwill be less thanbif and only if (X−μ)/σis less than (b−μ)/σ, and so
P{X<b}=P
{
X−μ
σ
<
b−μ
σ
}

(
b−μ
σ
)
Similarly, for anya<b,
P{a<X<b}=P
{
a−μ
σ
<
X−μ
σ
<
b−μ
σ
}
=P
{
a−μ
σ
<Z<
b−μ
σ
}
=P
{
Z<
b−μ
σ
}
−P
{
Z<
a−μ
σ
}

(
b−μ
σ
)
−
(
a−μ
σ
)