Robbins lemma

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

$\mathrm{E}(Xf(X-1))=\lambda \mathrm{E}(f(X)).$

Robbins introduced this proposition while developing empirical Bayes methods.