8 Incomplete Gamma and Related FunctionsApplications8.22 Mathematical Applications8.24 Physical Applications

The functions $P\left(a,x\right)$ and $Q\left(a,x\right)$ are used extensively in statistics as the probability integrals of the gamma distribution; see Johnson et al. (1994, pp. 337–414). Particular forms are the chi-square distribution functions; see Johnson et al. (1994, pp. 415–493). The function ${\mathrm{B}}_{x}\left(a,b\right)$ and its normalization ${I}_{x}\left(a,b\right)$ play a similar role in statistics in connection with the beta distribution; see Johnson et al. (1995, pp. 210–275). In queueing theory the Erlang loss function is used, which can be expressed in terms of the reciprocal of $Q\left(a,x\right)$; see Jagerman (1974) and Cooper (1981, pp. 80, 316–319).