The Standard Cauchy Distribution

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

Г(1)

VnT(1/2)(1 + x2)

 

1

n (1 + x2)’

 

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

 

1

n (1 + x2)

 

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

image261

image262

and its density is

Подпись: hm,n (x )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,

image264 Подпись: if n > 5, if n = 3, 4, if n = 1, 2. Подпись: (4.42)

= 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.

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