A different question is to infer the the Poisson
of the signal. Using once more Bayes theorem we get,
under the hypothesis of signal objects:

(45) |

Assuming a uniform prior for we get (see e.g. Ref. [2]):

(46) |

with expected value and variance both equal to and mode equal to (the expected value is shifted on the right side of the mode because the distribution is skewed to the right). Figure 12 shows these pdf's, for ranging from 0 to 12 and assuming a uniform prior for .

As far the pdf of that depends on all possible
values of , each with is probability, is concerned,
we get from probability theory
[and remembering that, indeed,
is equal to
, because depends only on
, and then the other way around]:

(47) |

i.e. the pdf of is the weighted average

The results for the example we are considering in this section are given in the plots of Fig. 11.