352 The Basics of financial economeTrics
F-Distribution
Our next distribution is the F-distribution. It is defined as follows. Let
Xn~(χ^21 ) and Yn~(χ^22 ).
Furthermore, assuming X and Y to be independent, then the ratio
Fnn =
X
n
Y
n(, 12 )^1
2(B.6)
has an F-distribution with n 1 and n 2 degrees of freedom inherited from the
underlying chi-square distributions of X and Y, respectively. We see that the
random variable in equation (B.6) assumes nonnegative values only because
neither X nor Y are ever negative. Hence, the support is on the nonnega-
tive real numbers. Also like the chi-square distribution, the F-distribution is
skewed to the right.
Once again, it is unnecessary to present the formula for the density
function. Figure B.6 displays the density function for various degrees of
freedom. As the degrees of freedom n 1 and n 2 increase, the function graph
becomes more peaked and less asymmetric while the tails lose mass.
FigURe B.6 Density Function of the F-Distribution for Various Degrees of Freedom
n 1 and n 2
(^00) 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
x
f(x
)
n 1 = 4, n 2 = 4
n 1 = 4, n 2 = 10
n 1 = 10, n 2 = 4
n 1 = 10, n 2 = 100