In statistics, the Robbins lemma, named after Herbert Robbins, states that if X is a random variable having a Poisson distribution with parameter λ, and f is any function for which the expected value E(f(X)) exists, then[1]
\operatorname{E}(Xf(X-1))=λ\operatorname{E}(f(X)).
Robbins introduced this proposition while developing empirical Bayes methods.