Including Gaussian uncertainty on the background estimate for upper limit calculations using Poissonian sampling

A procedure to include the uncertainty on the background estimate for upper limit calculations using Poissonian sampling is presented for the case where a Gaussian assumption on the uncertainty can be made. Under that hypothesis an analytic expression of the likelihood is derived which can be written in terms of polynomials defined by recursion. This expression may lead to a significant speed up of computing applications that extract the upper limits using Toy Monte Carlo.