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 \(t \ge 0\),
\[ \Gamma(t+1) \;\;=\;\; t! \]