# The Standard Cauchy Distribution

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

 h1(x)

 (4.39)

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:

 TO 2П f exp< — TO

 i ■ t ■ x)exp(—|t |)dt

 (4.40)

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 moment-generating 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.