# Theorem 24 suppose that y 1 y n are random variables

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Theorem 2.4.Suppose thatY1,...,Ynare random variables with jointpdf or pmff(y1,...,yn).Let{i1,...,ik} ⊂ {1,...,n},and letf(yi1,...,yik) bethe marginal pdf or pmf ofYi1,...,Yikwith supportYYi1,...,Yik.Assume thatE[h(Yi1,...,Yik)] exists. ThenE[h(Yi1,...,Yik)] =integraldisplay-∞· · ·integraldisplay-∞h(yi1,...,yik)f(yi1,...,yik)dyi1· · ·dyik=integraldisplay· · ·integraldisplayYYi1,...,Yikh(yi1,...,yik)f(yi1,...,yik)dyi1· · ·dyikiffis a pdf, andE[h(Yi1,...,Yik)] =summationdisplayyi1· · ·summationdisplayyikh(yi1,...,yik)f(yi1,...,yik)=summationdisplay(yi1,...,yik)∈YYi1,...,Yikh(yi1,...,yik)f(yi1,...,yik)iffis a pmf.Proof.The proof for a joint pdf is given below. For a joint pmf, replacethe integrals by appropriate sums.Letg(Y1,...,Yn) =h(Yi1,...,Yik).ThenE[g(Y)] =integraldisplay-∞· · ·integraldisplay-∞h(yi1,...,yik)f(y1,...,yn)dy1· · ·dyn=integraldisplay-∞· · ·integraldisplay-∞h(yi1,...,yik)bracketleftbiggintegraldisplay-∞· · ·integraldisplay-∞f(y1,...,yn)dyik+1· · ·dyinbracketrightbiggdyi1· · ·dyik=integraldisplay-∞· · ·integraldisplay-∞h(yi1,...,yik)f(yi1,...,yik)dyi1· · ·dyiksince the term in the brackets gives the marginal.QED
CHAPTER 2.MULTIVARIATE DISTRIBUTIONS42Example 2.5.TypicallyE(Yi),E(Y2i) andE(YiYj) forinegationslash=jare of pri-mary interest. Suppose that (Y1,Y2) has joint pdff(y1,y2).ThenE[h(Y1,Y2)]=integraldisplay-∞integraldisplay-∞h(y1,y2)f(y1,y2)dy2dy1=integraldisplay-∞integraldisplay-∞h(y1,y2)f(y1,y2)dy1dy2where-∞tocould be replaced by the limits of integration fordyi.Inparticular,E(Y1Y2) =integraldisplay-∞integraldisplay-∞y1y2f(y1,y2)dy2dy1=integraldisplay-∞integraldisplay-∞y1y2f(y1,y2)dy1dy2.Since finding the marginal pdf is usually easier than doing the doubleintegral, ifhis a function ofYibut not ofYj, find the marginal forYi:E[h(Y1)] =integraltext-∞integraltext-∞h(y1)f(y1,y2)dy2dy1=integraltext-∞h(y1)fY1(y1)dy1.Similarly,E[h(Y2)] =integraltext-∞h(y2)fY2(y2)dy2.In particular,E(Y1) =integraltext-∞y1fY1(y1)dy1,andE(Y2) =integraltext-∞y2fY2(y2)dy2.Suppose that (Y1,Y2) have a joint pmff(y1,y2).Then theexpectationE[h(Y1,Y2)] =y2y1h(y1,y2)f(y1,y2) =y1y2h(y1,y2)f(y1,y2).Inparticular,E[Y1Y2] =summationdisplayy1summationdisplayy2y1y2f(y1,y2).Since finding the marginal pmf is usually easier than doing the doublesummation, ifhis a function ofYibut not ofYj, find the marginal forpmf forYi:E[h(Y1)] =y2y1h(y1)f(y1,y2) =y1h(y1)fY1(y1).Similarly,E[h(Y2)] =y2h(y2)fY2(y2).In particular,E(Y1) =y1y1fY1(y1) andE(Y2) =y2y2fY2(y2).

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