Uncertainty on the expected background

In these examples we made the assumption that the expected number of background events is well known. If this is not the case, we can quantify our uncertainty about it by a pdf $f(\lambda_b)$, whose modeling depends on our best knowledge about $\lambda _s$. Taking account of this uncertainty in a probabilistic approach is rather simple, at least conceptually (calculations can be quite complicate, but this is a different question). In fact, applying probability theory we get:

$$f(p\,\vert\,x,\,n) = \int_0^\infty\!\! f(p\,\vert\,x,\,n,\,\lambda_b)\,f(\lambda_b) \,\mbox{d}\lambda_b\,.$$ (34)

We recognize in this formula that the pdf that takes into account all possible values of $\lambda$ is a weighted average of all $\lambda_b$ dependent pdf's, with a weight equal to $f(\lambda_b)$.