Page 361 - Applied Probability

Appendix B: The Normal
Distribution

B.1 Univariate Normal Random Variables

A random variable X is said to be standard normal if it possesses the
density function

1 x 2
ψ(x) = √ e − 2 .
2π
ˆ
To find the characteristic function ψ(s)= E(e isX )of X, we derive and
solve a differential equation. Differentiation under the integral sign and
integration by parts together imply that

d 1 ∞ isx x 2
ˆ
ψ(s) = √ e ixe − 2 dx
ds 2π −
i ∞ isx d x 2
= −√ e e − 2 dx
2π − dx
−i 2 ∞ s ∞ x 2
4
e
= √ e isx − x 4 4 − √ e isx − 2 dx
e
2
2π − 2π −
ˆ
= −sψ(s).
ˆ
The unique solution to this differential equation with initial value ψ(0) = 1
ˆ
2
is ψ(s)= e −s /2 . The differential equation also yields the moments
1 d
ˆ
E(X)= ψ(0)
i ds
=0
1 d 2
ˆ
2
E(X )= ψ(0)
2
i ds 2
1 2 ˆ
ˆ
= − ψ(s)+ s ψ(s)
i 2 s=0
=1.
An affine transformation Y = σX + µ of X is normally distributed with
density
1 y − µ 1 (y−µ) 2
ψ = √ e − 2σ 2 .
σ σ 2πσ