Continuous Probability Distributions Commonly Used in Financial Econometrics 347
values, n < 30 justifies the use of the standard normal distribution for the
left-hand side of equation (B.2).
These properties make the normal distribution the most popular dis-
tribution in finance. This popularity is somewhat contentious, however, for
reasons that will be given when we describe the α-stable distribution.
The last property we will discuss of the normal distribution that is
shared with some other distributions is the bell shape of the density func-
tion. This particular shape helps in roughly assessing the dispersion of the
distribution due to a rule of thumb commonly referred to as the empirical
rule. Due to this rule, we have
PX([∈μ±σ])=μFF()+σ−μ()−σ≈68%PX([∈μ±σ2])(=μFF+σ2)−μ(2−σ)≈95%PX([∈μ±σ3])(=μFF+σ3)−μ(3−σ) 100%≈The above states that approximately 68% of the probability is given
to values that lie in an interval one standard deviation σ about the mean
μ. About 95% probability is given to values within 2σ to the mean, while
nearly all probability is assigned to values within 3σ from the mean.
Chi-Square Distribution
Our next distribution is the chi-square distribution. Let Z be a standard
normal random variable, in brief Z ~ N (0,1), and let X = Z^2. Then X is
distributed chi-square with one degree of freedom. We denote this as X ~
χ^2 (1). The degrees of freedom indicate how many independently behaving
standard normal random variables the resulting variable is composed of.
Here X is just composed of one, namely Z, and therefore has one degree
of freedom.
Because Z is squared, the chi-square distributed random variable
assumes only nonnegative values; that is, the support is on the nonnegative
real numbers. It has mean E(X) = 1 and variance var(X) = 2.
In general, the chi-square distribution is characterized by the degrees
of freedom n, which assume the values 1, 2,... , and so on. Let X 1 , X 2 ,... ,
Xn be n χ^2 (1) distributed random variables that are all independent of each
other. Then their sum, S, is
=χ∑
SXi~(n)
in
2
1