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
E ( X f ( X − 1 ) ) = λ E ( f ( X ) ) . {\displaystyle \operatorname {E} (Xf(X-1))=\lambda \operatorname {E} (f(X)).}
Robbins introduced this proposition while developing empirical Bayes methods.
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