The Cauchy distribution has density function f (x) = 1 (1 + x2) for x R. (5.39) The gamma distribution with parameters w (>

The Cauchy distribution has density function f (x) =

1

π(1 + x2)

for x ∈ R. (5.39)

The gamma distribution with parameters w (> 0) and λ (> 0) has density function f (x) =





1

Ŵ(w)

λwxw−1e−λx if x > 0, 0 if x ≤ 0,

(5.40)

where Ŵ(w) is the gamma function, defined by

Ŵ(w) =

Z

∞

0 xw−1e−x dx. (5.41)

Note that, for positive integers w, Ŵ(w) = (w − 1)! (see Exercise 5.46).

The beta distribution with parameters s, t (> 0) has density function f (x) =

1 B(s, t)

xs−1(1 − x)t−1 for 0 ≤ x ≤ 1. (5.42)

The beta function B(s, t) =

Z 1 0

xs−1(1 − x)t−1 dx (5.43)

is chosen so that f has integral equal to one. You may care to prove that B(s, t) =

Ŵ(s)Ŵ(t)

Ŵ(s + t)

(see (6.44)). If s = t = 1, then X is uniform on [0, 1].

The chi-squared distribution with n degrees of freedom(sometimes written χ2 n ) has density function f (x) =





1 2Ŵ( 1 2n)

( 1 2 x)

1 2 n−1e−1 2 x if x > 0, 0 if x ≤ 0.

(5.44)