  Suppose that $X_1,X_2,\ldots$ are a stream of independent, identically
distributed Poisson random variables with mean $\mu$. This work presents a new
estimate $\mu_k$ for $\mu$ with the property that the distribution of the
relative error in the estimate ($(\hat \mu_k/\mu) - 1$) is known, and does not
depend on $\mu$ in any way. This enables the construction of simple exact
confidence intervals for the estimate, as well as a means of obtaining fast
approximation algorithms for high dimensional integration using TPA. The new
estimate requires a random number of Poisson draws, and so is best suited to
Monte Carlo applications. As an example of such an application, the method is
applied to obtain an exact confidence interval for the normalizing constant of
the Ising model.
