The Standard Cauchy Distribution
The ti distribution is also known as the standard Cauchy distribution. Its density is






where the second equality follows from (4.36), and its characteristic function is
Vh(t) = exp(t ).
The latter follows from the inversion formula for characteristic functions:







See Appendix 4.A. Moreover, it is easy to verify from (4.39) that the expectation of the Cauchy distribution does not exist and that the second moment is infinite.
4.6.2. The F Distribution
Let Xm ~ x2 and Yn ~ x2, where Xm and Yn are independent. Then the distribution of the random variable
F _ Xm / m Yn/n
is said to be F with m and n degrees of freedom and is denoted by Fm, n. Its distribution function is
and its density is
mm/2 Y(m/2 + n/2) xm/2—1
nm/2 T(m/2)T(n/2) [1 + m ■ x/n]m/2+n/2 ’
See Appendix 4.A.
Moreover, it is shown in Appendix 4.A that
E[F] = n/(n — 2) if n > 3,
= to if n = 1, 2,
Furthermore, the momentgenerating function of the Fm, n distribution does not exist, and the computation of the characteristic function is too tedious an exercise and is therefore omitted.
Leave a reply