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Defineafunctionk(x,y) h(x)/h(y) = 1, whichisboundedandnonzero for any x ∈Xand y ∈X Note that x and y such that n i=1 x i = n i=1 y i are equivalent because function k(x,y) satisfies the requirement of likelihood ratio partition Therefore, T(x) n i=1 x i is a sufficient statistic Problem 5 Let X1,X2,,X m and Y1,Y2,,Y n be two independent sam ples from N(µ,σ2)andN(µ,τ27 9 = 4 5 $ ½ j ½ $ c $ f b $ f g 9 > e t w , 2 x , $ Ç Ç Ç Ç Ç Ç Ç Ç Ç Ç Ç Ç Ç Ç Ç Ç Ç Ç Ç Ç Ç Ç Ç Ç Ç Ç , ª s $ 7 $ o f 5 4 e $ j 5 4 i > e 9 = $ $ j ½ $ c $ ­ / 0 ¢ ÈX z ú w å Á Â { ^ t z ¹ å Í É ç x Z Â å p b { ú ` o S M M h i Z b { *9* s E « § Ù Ä

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Ö ç F Ö ¤ R 7 Ö ¥ ) ç F v 2VDND 0HWUR I W µ Ä ¡ Î y ã à ) b q O d } ´ I W µ Ä ¡ Î µ à ê ¸ ¸ L z ¯ b d } ´2VDND 0HWUR I W µ Ä ¡ Î Ê Y µProof As we know, X and Y are independent if and only if fX;Y(x;y) = fX(x)fY(y) or, equivalently, fXjY(xjy)= fX(x) But then E(XjY =y)=åx xfXjY(xjy)=åx xfX(x)=E(X) 2 2 EE(g(X)jY)=E(g(X)) Proof Set Z = g(X) Statement (i) of Theorem 1 applies to any two rv's Hence, applying it to Z and Y we obtain EE(ZjY)= E(Z) which is the same Calculus Basic Differentiation Rules Quotient Rule 1 Answer Noah G Start by simplifying the entire expression f (x) = x x2c x f (x) = x2 x2 c f '(x) = 2x(x2 c) −2x(x2) (x2 c)2 f '(x) = 2x3 2xc − 2x3 (x2 c)2

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But avoid Asking for help, clarification, or responding to other answersLinks with this icon indicate that you are leaving the CDC website The Centers for Disease Control and Prevention (CDC) cannot attest to the accuracy of a nonfederal website Linking to a nonfederal website does not constitute an endorsement by CDC or any of its employees of the sponsors or the information and products presented on the website" #" " #" E(g(X,Y))=g(x,y)f XY (x,y) It is important to note that if the function g(x,y) is only dependent on either x or y the formula above reverts to the 1dimensional case Ex Suppose X and Y have a joint pdf f XY(x,y) Calculate E(X

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