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{{citation|title=Stochastic Modeling and Mathematical Statistics: A Text for Statisticians and Quantitative Scientists|first=Francisco J.|last=Samaniego|publisher=CRC Press|year=2015|isbn=9781466560475|page=118|url=https://books.google.com/books?id=v1HSBQAAQBAJ&pg=PA118}}.

: $\backslash operatorname\{E\}(X\; f(X\; -\; 1))\; =\; \backslash lambda\; \backslash operatorname\{E\}(f(X)).$

Robbins introduced this proposition while developing empirical Bayes methods.