2.4 The Incomplete Gamma and Beta Functions
The gamma and beta functions were defined by definite integrals over their full ranges. Cutting the range short gives two further functions, and these are the ones that carry the classical distributions of statistics.
Definition 2.4.1 (Incomplete gamma functions). \[\gamma (s,x) = \int ^{x}_{0}t^{s-1}e^{-t}\,dt, \qquad \Gamma (s,x) = \int ^{\infty }_{x}t^{s-1}e^{-t}\,dt,\] the lower and upper incomplete gamma functions, so that \(\gamma (s,x)+\Gamma (s,x)=\Gamma (s)\). The regularised form is \(P(s,x) = \gamma (s,x)/\Gamma (s)\).
Definition 2.4.2 (Incomplete beta function). \[\beta _x(a,b) = \int ^{x}_{0}t^{a-1}(1-t)^{b-1}\,dt, \qquad 0\leq x\leq 1,\] with regularised form \(I_x(a,b) = \beta _x(a,b)/\beta (a,b)\).
Proof. Integrate \(\gamma (s+1,x)=\int ^{x}_{0}t^{s}e^{-t}dt\) by parts, taking \(u=t^{s}\) and \(dv=e^{-t}dt\): \[\gamma (s+1,x) = \left [-t^{s}e^{-t}\right ]^{x}_{0} + s\int ^{x}_{0}t^{s-1}e^{-t}\,dt = -x^{s}e^{-x} + s\,\gamma (s,x).\] Letting \(x\to \infty \) recovers \(\Gamma (s+1)=s\,\Gamma (s)\), the functional equation of Section 2.1. □
Note 2.4.4. These two functions are the distribution functions of statistics written in analytic form, and the correspondence is worth setting out explicitly.
| Distribution | Distribution function |
| Gamma\((s,\theta )\) | \(P\left (s,\,x/\theta \right )\) |
| Chi-square on \(k\) degrees of freedom | \(P\left (k/2,\,x/2\right )\) |
| Poisson\((\lambda )\), \(P(X\leq k)\) | \(\Gamma (k+1,\lambda )/k!\) |
| Beta\((a,b)\) | \(I_x(a,b)\) |
| Binomial\((n,p)\), \(P(X\leq k)\) | \(I_{1-p}(n-k,\,k+1)\) |
| Student’s \(t\) on \(\nu \) degrees of freedom | \(I\) evaluated at \(\nu /(\nu +t^{2})\) |
| \(F\) on \((d_1,d_2)\) degrees of freedom | \(I\) evaluated at \(d_1x/(d_1x+d_2)\) |
Every entry in the right-hand column is one of the two functions defined above. This is why a statistical table is possible at all: the whole collection reduces to two functions of two arguments, and tabulating those tabulates everything. It is also why the chi-square and the gamma share a distribution function, and why the \(t\) and \(F\) distributions — which look quite unrelated — are evaluated by the same routine in every statistical library.
Example 2.4.5. For integer \(s=n\), repeated application of the recurrence gives \[\Gamma (n,x) = (n-1)!\,e^{-x}\sum ^{n-1}_{k=0}\frac {x^{k}}{k!} .\] Dividing by \((n-1)!\) shows that the upper incomplete gamma function at integer argument is exactly the Poisson distribution function, which is the third row of Table 2.1. The identity is the analytic statement of a fact familiar from probability: the waiting time to the \(n\)th event of a Poisson process exceeds \(x\) precisely when fewer than \(n\) events have occurred by \(x\).
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