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, then¹

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

Robbins introduced this proposition while developing empirical Bayes methods.