Mathematical interlude: Gamma functions

Various results relating to Poisson processes can be derived most easily using a particular kind of mathematical function called a gamma function. In particular, the mean and variance of the exponential distribution are easiest to find using gamma functions.

Definition

The gamma function has a single argument and is defined by

\[ \Gamma(t) \;\;=\;\; \int_0^{\infty} {x^{t-1} e^{-x}} \; dx \]

Gamma functions have various useful properties.

Recursive formula

For any \(t\),

\[ \Gamma(t+1) \;\;=\;\; t \times \Gamma(t) \]

The proof of this result requires integration by parts.

\[ \begin{align} \Gamma(t+1) \;\;&=\;\; \int_0^{\infty} {x^t e^{-x}} \; dx \\ &=\;\; \int_0^{\infty} {tx^{t-1} e^{-x}} \; dx + \left[x^t \times -e^{-x} \right]_{x=0}^{\infty} \\ &=\;\; t \times \Gamma(t) \end{align} \]

Note that \(\left[x^t \times -e^{-x} \right]\) is zero for both \(x = 0\) and \(x = \infty\).

Two specific values of the gamma function are now given.

Two specific values

\[ \Gamma(1) \;\;=\;\; 1 \spaced{and} \Gamma \left(\frac 1 2\right) = \sqrt{\pi} \]

The first of these is easily proved by integration. The value of \(\Gamma(\frac 1 2)\) is much harder to derive and the proof is not given here.

Gamma functions can be treated as a generalisation of factorials. The following result shows their relationship.

Relationship to factorials

For any integer \(t \ge 0\),

\[ \Gamma(t+1) \;\;=\;\; t! \]
\[ \begin{align} \Gamma(t+1) \;\;&=\;\; t \times \Gamma(t) \\ &=\;\; t(t-1) \times \Gamma(t-1) \\ &=\;\; t(t-1)\cdots 1 \times \Gamma(1) \\ &=\;\; t! \end{align} \]