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X ismultivariatenormal⇔ a′x isnormalforalla def'n x ∼ Np(µ,Σ) ⇔ a′x ∼ N(a′µ,a′p thm If x ∼ Np(µ,Σ) then its characteristic function is φx(t) = exp(it′µ− 1 2t ′Σt) Proof Let y = t′x Then 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 xDefinition 2 Let X and Y be random variables with their expectations µ X = E(X) and µ Y = E(Y), and k be a positive integer 1 The kth moment of X is defined as E(Xk) If k = 1, it equals the expectation 2 The kth central moment of X is defined as E(X − µ X)k If k = 2, then it is called the variance of X and is denoted by var(X)Title Microsoft Word 2122·üºóncÀ'¤óÕë¨ó¶ï¯Áón¥k¢Y ã ³Ôüdocx Author rtamura Created Date PM

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10 COV(X,Y) = E(X E(X)) * (Y E(Y) = E(XY) E(X)E(Y) Question What is COV(X,X)?DZIZUZ V r P E r O pr@NSQMPE_AJDA}v @uywysz o i m h F Y C P G r T J efcJYjULlVW SQLRMFjcHJNXK x s t t {` _ G r T J mikhosLnBKW abBOWMF ^ f k q d l p g e OPROFC\qG^dA x ` t X ug T H b Z w ` a I E x s t t {A n \ SG O $ y a X i ^ M { Ó ( x Í $ Ó Ò å ­ ¿ Ä x Ç V b { y w O t s b { "$ 7 ¯ ï ï Ä ¢ ¯ ï ï Ä A q b ¨ µ ¯ ï é w Ô ù w £ ¨ µ ¢ £ l Ó y º G O x ¨ µ ¯ ï é ) " 3), p w G O p b { y C ä ;

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Distributions Derived from Normal Random Variables χ 2 , t, and F Distributions Statistics from Normal Samples Normal Distribution Definition A Normal / Gaussian random variable X ∼ N(µ, σB1 C Y ¹ µ ñ ® Ð Ç Ì î D N 4 « ç g 4 k ± ¹ µ ù * O G I è r ì ð X ð H ç k x 4 i ¥ 9 ÷ ¯ é û ¶ ¹ µ / !Prove that the norm k·kX is induced by a scalar product, and thus X is a Hilbert space Show that {xn}∞ n=1 must then be an orthonormal sequence Solution We denote by S the linear span of {xn}∞ n=1 (the set of finite linear combinations of elements in {xn}∞ n=1)By property (b), we find that on S the norm kkX coincides with the ℓ2norm of its coefficients

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@ ô ì f Ä \ G ³ ( ` µ » ¾ p Û ) ô ` * G (, e Ï Û ) õ ( É ' Ö Û ) õ ( n Å Û ) ;E 5 V W X ž 6 J g n o Ü ⁄ J @ ¥ p ˘ « q — x p ˘ « r s µ ‡ X r ‡ X V x ˆ p ˇ ˆ « c < y t 3 L u v w B µ x y 5For a given function g and a specific value of θ, suppose that g0(θ) exists and is not 0Then √ ng(Yn)−g(θ) → N(0,σ2g0(θ)2) in distribution Proof The Taylor expansion of g(Yn) around Yn = θ is g(Yn) = g(θ)g0(θ)(Yn −θ)remainder, where the remainder→ 0 as Yn → θSince Yn → θ in probability it follows that the remainder→ 0 in probability By applying Slutsky

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Where rv's X(n) j are independent of each other and have the same distribution as a given integervalued rv X Theorem 2 can be used in order to prove the following statements Suppose that E(X)=µ, Var(X)=s2 Then (i) E(Yn)=µn (ii)If µ 6= 1, then Var(Yn)= s2µn¡1(1¡µn) (1¡µ) If µ =1 then Var(Yn)=ns 2Title ILS pdf Author tinal Created Date Z1, E(Y)=µ2, Var(X)=s2 2 2 Xj(Y = y) is a normal rv To verify this statement we substitute the necessary ingredients Hence, from the uniqueness of the joint mgf, Y » N(Amb;AVAT) Note that from (2) a subset of the Y0s is multivariate normal NOTE The results concerning the vector of means and variancecovariance matrix for linear

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µ θσ 2 /2 1 = EX = Ee Y = M Y (1) = e µ 2θ2σ 2 2 = EX 2 = Ee 2Y = M Y (2) = e (b) First, note that µ 2 σ 2 2 /(µ 1) = e It follows that a methodofmoments estimate for σ 2 is σˆ 2 = ln(ˆµ 2 /µˆ 2 1) where µˆ 1 = 1 n X i n i =1 µˆ 2 = 1 n X 2 nB 3 { ô N b È V £ ö J ä û $ Q C v g { ¼ § % Q L b À ¼ § & ¼ § ½ < n Ô ã Q l ' ¼ § » ã Q l $ ½ o L % L ï & D Î L ' û5ß ² L H s } 9 ô X j µ Í Ø n ê m K { Ð < Õ û $ % & ' È V £ Ú µ Í Ø n C v g { n M è û $ ¼ § B 3 { % C v & õ Ü B y LY n m Í o%3, Þ è s b q O d } j Ü ê µ r z I T y n m Í o ¦ " / g s b q O d } ¢ ñ Ç r z È ¢ ñ Ç < y T L r M Þ é ª Á Ï ® é Õ s y û Æ Â ê c q y n m È V b j è Ä é ¤ Ö y U q O d } Ö ' Ï Æ È )  s y Ì s b q z ¶ y ¢ ñ Ó ç V

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