Prove the following analogs to Stein's Lemma, assuming appropriate conditions on the function
g
. (a) If
x∼\gamma (\alpha ,\beta )
, then
E(g(x)(x-\alpha \beta ))=\beta E(xg^(')(x)).
(b) If
x∼\beta (\alpha ,\beta )
, then
E[g(x)(\beta -(\alpha -1)((1-x))/(x))]=E((1-x)g^(')(x))

Question

Prove the following analogs to Stein's Lemma, assuming appropriate conditions on the function

g

. (a) If

x∼\gamma (\alpha ,\beta )

, then

E(g(x)(x-\alpha \beta ))=\beta E(xg^(')(x)).

(b) If

x∼\beta (\alpha ,\beta )

, then

E[g(x)(\beta -(\alpha -1)((1-x))/(x))]=E((1-x)g^(')(x))

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