Anon

Continuous Probability Distributions Commonly Used in Financial Econometrics 345

A problem is that the distribution function cannot be solved for analyti-
cally and therefore has to be approximated numerically. In the particular
case of the standard normal distribution, the values are tabulated. Standard
statistical software provides the values for the standard normal distribution
as well as most of the distributions presented in this chapter. The standard
normal distribution is commonly denoted by the Greek letter Φ such that
we have Φ=()()xFxP=≤()Xx, for some standard normal random variable
X. In Figure B.2, graphs of the distribution function are given for three dif-
ferent sets of parameters.

properties of the normal distribution

The normal distribution provides one of the most important classes of prob-
ability distributions due to two appealing properties:

Property 1. The distribution is location-scale invariant. That is, if X has

a normal distribution, then for every constant a and b, aX + b is

again a normal random variable.

FigURe B.2 Normal Distribution Function for Various Parameter Values

−3 −2 −1 0 1 2 3

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

Standard

normal

distribution

F(

x)

μ = 0, σ = 1

μ = 0, σ = 0.5

μ = 0, σ = 2