D.1 Derivation of p(sN _on, I)

In Section 14.4, we explored a Bayesian analysis of ON/OFF measurements, where ON is signal + background and OFF is a just the background. The background is only known imprecisely from OFF measurement. In this appendix, we derive Equation (14.17) for p( s N _on, I), the posterior probability of the signal event rate.

Our starting point is Equation (14.16), which we repeat here together with some of the other relevant equations:

(D.1)

(D.2)

(D.3)

(D.4)

The denominator of p( s, b N _on, I) in Equation (D.2) is given by

(D.5)

Substituting Equations (D.5), (D.4), (D.3) and (D.2) into Equation (D.1), we obtain

(D.6)

D.1.1 Evaluation of Num

We start with a binomial expansion of ( s + b) ^{N on}.

(D.7)

The numerator of Equation (D.6) becomes

(D.8)

We now want to evaluate the integral in Equation (D.8), which we first rewrite in the form of an incomplete gamma function:

(D.9)

Compare this to one form of the incomplete gamma function:

(D.10)

Thus, Equation (D.9) can be rewritten as

(D.11)

Provided b _max[ T _on + T _off] ? [ N _on + N _off ? i], we have that

(D.12)

Substituting Equation (D.12) into Equation (D.11), we obtain

(D.13)

Now substitute Equation (D.13) into Equation (D.8) to obtain

(D.14)