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E(XYjY =y)=E(yXjY =y)=yE(XjY =y)(because y is a constant) Hence, E(XYjY)= YE(XjY) by the definition of the conditional expectation 2 Corollary E(XY)=EYE(XjY) Proof E(XY)=EE(XYjY)= EYE(XjY) 2 Exercise Use the same method to prove that E(Xh(Y)jY) = h(Y)E(XjY) for any real valued function h(y)Ç ® nª¥ µ ¦µ µ¦ ¦´ ª·µ® µ ¼ nµ ¦³Á « ¨³° r µ¦ ¦³®ªnµ ¦³Á « ¸É ´Ê °¥¼nÄ ¦³Á «Å ¥ µ ´ µ¦Á ®¤µ¥ ¹ µ ´ µ¦Á ®¦º° » ¨ ¸0, if x < 0 This is derived via computing d dx F(x) for where Θ(x) denotes the cdf of N(0,1) Observing that E(X) = E(eY) and E(X2) = E(e2Y) are simply the moment generating function (MGF) M Y (s) = E(esY) of Y ∼ N(µ,σ2) evaluated at s = 1 and s = 2 respectively yields E(X) = eµσ 2 2 E(X2) = e2µ
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Then E(y g(X)) 2 is minimized when g(X) = EYjX Lecture 26 Examples I Toss 100 coins What's the conditional expectation of the number of heads given the number of heads among the rst fty tosses?KD } v } o E µ u í ô ð ì r ì ô ð õ Æ ï l ï í l î ì î ð í Y µ o Ç µ P v Æ v µ Z } v P ( } , Z& / U // U v /// ~ ~ í / v µ } v o W } } v U ~ ~ î U v ~ ~ ï U ( o oDefinition via Cram´erWold x ismultivariatenormal⇔ a′x isnormalforalla thm If x ∼ Np(µ,Σ) then its characteristic function is φx(t) = exp(it′µ− 1 2t ′Σt) Proof Let y = t′xThen the cf of y is φy(s) def= E{eisy} = exp{isE(y)−1 2s 2var(y)} = exp{ist′µ−1 2s 2t′Σt} Then the cf of x is
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