Mathematical interlude

Various results relating to Poisson processes can be derived most easily using a particular kind of mathematical function called a gamma function.

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 \(t\),

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

Two specific values

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

Relationship to factorials

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

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

(All proved in full version)