XMM-Newton Science Analysis System

## Null-hypothesis probability distribution for a weighted sum of Poissonian variates

The approach followed in the present section is essentially that of Fay and Feuer [1]. See also Stewart []

Let be a random deviate which follows a Poissonian probability distribution about the expectation value . Although the Poisson distribution itself is only defined for integer , one can find the following continuous `envelope function' to the Poisson values:

(3)

where is the gamma function

Note that both the expectation value and variance of the function are identical to those of the corresponding discrete Poisson distribution, namely both equal to . Now, given random deviates , each of which follows a distinct Poisson distribution about its average , let us form the weighted sum

The expectation value of is

the variance can in similar fashion be shown to equal

The probability function only has values where all the are integer and generally speaking may be expected to be a messy-looking and intractable function. However, recall that for purposes of source detection we are not interested in but in the integral of from a particular sample of to infinity. is stepwise continuous, the steps becoming smaller and denser as increases. The envelope function with the same expectation value and variance as is given by

(4)

where

In plain English, what comparison of equations 3 and 4 suggests is that a weighted sum of Poisson variates behaves approximately like a single Poisson distribution with the same average and variance. Therefore since, for a single Poisson variate , the probability for to be greater than some sampled value is given by

(where is the incomplete gamma function we met with in section 3.2) we postulate that the equivalent expression for is approximately given as follows:

(5)

Equation 5 is used both in the present program and in boxdetect to estimate the null-hypothesis probability distribution of the weighted sum of Poissonian images.

XMM-Newton SOC -- 2021-11-30