Anon

344 The Basics of financial economeTrics

Normal Distribution

The first distribution we discuss is the normal distribution. It is the distri-
bution most commonly used in finance despite its many limitations. This
distribution, also referred to as the Gaussian distribution, is characterized by
the two parameters: mean (μ) and standard deviation (σ). The distribution
is denoted by N(μ, σ^2 ). When μ = 0 and σ^2 = 1, then we obtain the standard
normal distribution.
The density function for the normal distribution is given by

(^) fx e
x
()= ⋅

1 −()−

2

2

22

πσ

μ

σ (B.1)

The density function is symmetric about μ. A plot of the density function
for several parameter values is given in Figure B.1. As can be seen, the
value of μ results in a horizontal shift from 0 while σ inflates or deflates
the graph. A characteristic of the normal distribution is that the densities
are bell shaped.

FigURe B.1 Normal Density 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

x

μ = 0, σ = 1^

μ = 1, σ = 1

μ = 0, σ = 0.5^

μ = 0, σ = 2

Standard

normal

distribution

f(x

)