348 The Basics of financial economeTrics
In words, the sum is again distributed chi-square but this time with n
degrees of freedom. The corresponding mean is E(X) = n, and the variance
equals var(X) = 2 · n. So, the mean and variance are directly related to the
degrees of freedom.
From the relationship in equation (B.3), we see that the degrees of free-
dom equal the number of independent χ^2 (1) distributed Xi in the sum. If
we have two independent random variables X 1 ~ χ^2 (n 1 ) and X 2 ~ χ^2 (n 2 ), it
follows that
(^) XX 12 +χ~(^2 nn 12 + ) (B.4)
From equation (B.4), we have that chi-square distributions have Prop-
erty 2; that is, they are stable under summation in the sense that the sum
of any two independent chi-squared distributed random variables is itself
chi-square distributed.
We won’t present the chi-squared distribution’s density function here.
However, Figure B.3 shows a few examples of the plot of the chi-square
density function with varying degrees of freedom. As can be observed, the
chi-square distribution is skewed to the right.
FigURe B.3 Density Functions of Chi-Square Distributions for Various Degrees of
Freedom n
0 5 10 15
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
x
n = 1
n = 2
n = 5
n = 10
f(x
)