SUMMARY Consider a random sample, x1, …, xn, from a Poisson distribution. The Rao-Blackwell theorem and the completeness property of this distribution are exploited to show that E(kj\σxi = X) = x¯ (i = 1, 2, …) and to find higher conditional moments of the sample cumulants. Some possible applications to the testing of the fit of the Poisson distribution are suggested.