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)