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Continuous Probability Distributions Commonly Used in Financial Econometrics 349

Student’s t-Distribution

An important continuous probability distribution when the population
variance of a distribution is unknown is the Student’s t-distribution (also
referred to as the t-distribution and Student’s distribution).
To derive the distribution, let X be distributed standard normal, that is,
X ~ N(0,1), and S be chi-square distributed with n degrees of freedom, that
is, S ~ χ^2 (n). Furthermore, if X and S are independent of each other, then

Z

X

Sn

= tn

/

~() (B.5)

In words, equation (B.5) states that the resulting random variable Z is Stu-
dent’s t-distributed with n degrees of freedom. The degrees of freedom are
inherited from the chi-square distribution of S.
Here is how we can interpret equation (B.5). Suppose we have a popula-
tion of normally distributed values with zero mean. The corresponding nor-
mal random variable may be denoted as X. If one also knows the standard
deviation of X,

σ= var()X

with X/σ, we obtain a standard normal random variable.
However, if σ is not known, we instead have to use, for example,

Sn/1=⋅/(nX 122 ++... Xn)^

where XX 122 ,,... n are n random variables identically distributed as X.
Moreover, X 1 ,... , Xn have to assume values independently of each other.
Then, the distribution of

XS//n

is the t-distribution with n degrees of freedom, that is,

XS//nt~(n)^

By dividing by σ or S/n, we generate rescaled random variables that
follow a standardized distribution. Quantities similar to XS//n play an
important role in parameter estimation.
It is unnecessary to provide the complicated formula for the Student’s
t-distribution’s density function here. Basically, the density function of the