# Stein's lemma

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 Artikel ieu keur dikeureuyeuh, ditarjamahkeun tina basa Inggris. Bantosanna diantos kanggo narjamahkeun.

Stein's lemma, ngaran keur ngahargaan ka Charles Stein, may be characterized as a theorem of probability theory that is of interest primarily because of its application to statistical inference -- in particular, its application to James-Stein estimation and empirical Bayes methods.

## Statement of the lemma

Suppose X is a normally distributed random variable with nilai ekspektasi μ and varian σ2. Further suppose g is a function for which the two expectations E( g(X) (X − μ) ) and E( g ′(X) ) both exist (the existence of the expectation of any random variable is equivalent to the finiteness of the expectation of its absolute value). Then

$E(g(X)(X-\mu))=\sigma^2 E(g'(X)).$

In order to prove this lemma, recall that the probability density function for the normal distribution with expectation 0 and variance 1 is

$\varphi(x)={1 \over \sqrt{2\pi}}e^{-x^2/2}$

and that for a normal distribution with expectation μ and varian σ2 is

${1\over\sigma}\varphi\left({x-\mu \over \sigma}\right).$

Then use integration by parts.