Poisson background on the observed number of `trials' and of `successes'

- : the total observed numbers of `objects', of which are due to signal and to background; but these two numbers are not directly observable and can only be inferred;
- : the total observed numbers of the `objects' of the subclass of interest, sum of the unobservable and ;
- : the expected number of background objects;
- : the expected proportion of successes due to the background events.

Let us first calculate the probability function
that depends on the unobservable and .
This is the probability function
of the sum of two binomial variables:

where ranges between and , and ranges between and . can vary between 0 and , has expected value and variance . As for Eq. (32), we need to evaluate Eq. (35) only for the observed number of successes. Contrary to the implicit convention within this paper to use the same symbol meaning different probability functions and pdf's, we name Eq. (35) for later convenience.

In order to obtain the general likelihood we need, two observations are in order:

- Since depends from only via , then is equal to .
- The likelihood that depends also on can obtained from

by the following reasoning:- if , then

- else

- if , then

(36) | |||

(37) |

At this point we get rid of in the conditions, taking
account its possible values and their probabilities, given :

i.e.

where ranges between 0 and , due to the condition. Finally, we can use Eq. (39) in Bayes theorem to infer and :

We give now some numerical examples. For simplicity (and because we are not thinking to a specific physical case) we take uniform priors, i.e. . We refer to section 3.1 for an extensive discussion on prior and on critical `frontier' cases.