Anon

Continuous Probability Distributions Commonly Used in Financial Econometrics 351

dispersion from S relative to the standard normal distribution. By includ-
ing more independent N(0,1) random variables Xi such that the degrees
of freedom increase, S becomes less dispersed. Thus, much uncertainty
relative to the standard normal distribution stemming from the denomina-
tor in XS//n vanishes. The share of randomness in XS//n originating
from X alone prevails such that the normal characteristics preponderate.
Finally, as n goes to infinity, we have something that is nearly standard nor-
mally distributed.
The mean of the Student’s t random variable is zero, that is E(X) = 0,
while the variance is a function of the degrees of freedom n as follows

σ^2

2

==

−

var()X

n

n

For n = 1 and 2, there is no finite variance. Distributions with such small
degrees of freedom generate extreme movements quite frequently relative to
higher degrees of freedom. Precisely for this reason, stock price returns are
often found to be modeled quite well using distributions with small degrees
of freedom, or alternatively, distributions with heavy tails with power decay,
with power parameter less than 6.

FigURe B.5 Distribution Function of the t-Distribution for Various Degrees of Free-
dom n Compared to the Standard Normal Density Function N(0,1)

−5^0 −4 −3 −2 −1 0 1 2 3 4 5

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

F(

x)

t(1)^

t(5)

N(0,1)