Page 272 - Elements of Distribution Theory

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052184472Xc09 CUNY148/Severini June 2, 2005 12:8

258 Approximation of Integrals

Theorem 9.1. (x + 1) = x (x),x > 0, and for n = 0, 1,..., (n + 1) = n!.

Proof. Using integration-by-parts,

∞ ∞ ∞
x
x

(x + 1) = t exp(−t) dt =−t exp(−t) + x t x−1 exp(−t) dt = x (x).
0 0 0
Hence, for an integer n,
(n + 1) = n (n) = n(n − 1) (n − 1) =· · · = n! (1);
the second result now follows from the easily verified fact that (1) = 1.
A function closely related to the gamma function is the beta function. Let r and s be
nonnegative real numbers. Define
1
β(r, s) = t r−1 (1 − t) s−1 dt;
0
this function appears in the normalizing constant of the density of the beta distribution. The
following theorem gives an expression for the beta function in terms of the gamma function.

Theorem 9.2.
(r) (s)
β(r, s) = , r > 0, s > 0.
(r + s)
Proof. Note that, for r > 0 and s > 0,
∞ ∞

(r) (s) = t r−1 exp(−t) dt t s−1 exp(−t) dt
0 0
∞ ∞

t
= t r−1 s−1 exp{−(t 1 + t 2 )} dt 1 dt 2 .
1 2
0 0
Using the change-of-variable x 1 = t 1 + t 2 , x 2 = t 1 /(t 1 + t 2 ), we may write
∞ ∞

r−1 s−1
t 1 t 2 exp{−(t 1 + t 2 )} dt 1 dt 2
0 0
1 ∞
x
= x r+s−1 r−1 (1 − x 2 ) s−1 exp(−x 1 ) dx 1 dx 2
1 2
0 0
1 ∞
r−1 s−1 r+s−1
= x 2 (1 − x 2 ) dx 2 x 1 exp(−x 1 ) dx 1
0 0
= β(r, s) (r + s),
proving the result.
The value of the gamma function increases very rapidly as the argument of the function
increases. Hence, it is often more convenient to work with the log of the gamma function
rather than with the gamma function itself; a plot of this function is given in Figure 9.1.
Clearly, log (x) satisfies the recursive relationship
log (x + 1) = log x + log (x).
In addition, the following result shows that log (x)isa convex function.

Theorem 9.3. The function log (x),x > 0,isconvex.