Incorporation of the statistical uncertainty in the background estimate into the upper limit on the signal

We present a procedure for calculating an upper limit on the number of signal events which incorporates the Poisson uncertainty in the background, estimated from control regions of one or two dimensions. For small number of signal events, the upper limit obtained is more stringent than that extracted without including the Poisson uncertainty. This trend continues until the number of background events is comparable with the signal. When the number of background events is comparable or larger than the signal, the upper limit obtained is less stringent than that extracted without including the Poisson uncertainty. It is therefore important to incorporate the Poisson uncertainty into the upper limit; otherwise the upper limit obtained could be too stringent.