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Definition 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 isV v\qgurph iru sdwlhqwv lq wkh 86 dqg dsdq gdwd iurp D ODUJH WKUHH FHQWHU FRKRUW VWXG\ 0RG 5KHXPDWRO ,GHJXFKL 6XGD $ 7DNHQR 0 8HGD $ 2KQR 6 ,VKLJDWVXER < %HKFHW GLVHDVH HYROXWLRQ RI FOLQLFDO PDQLIHVWDWLRQV 0HGLFLQHLet X and Y be two discrete rv's with a joint pmf fX;Y(x;y) = P(X = x;Y = y) Remember that the distributions (or the pmf's) fX(x) = P(X = x) of X and fY(y) = P(Y = y) of Y are called the marginal distributions of the pare (X;Y) and that E(Yn)=µn (ii)If µ 6= 1, then Var
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Polyoxometalates In Solution Speciation Under Spotlight Chemical Society Reviews Rsc Publishing Doi 10 1039 D0csa
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, } o Ç / v v D o } µ v rs } v ( v v ô î õ ô E } Z t l Z u Z } D o } µ v &> ï î õ ð ì > Y µ v Ç t Ç v Z u D o } µ v r W o u Ç ð ñ í ì t E Á , À v À v µ D o } µ v &> ï î õ ì ðEskp(k) in the discrete rv case It is so called because it generates the moments of X by differentiation at s = 0 M0(0) = E(X), (7) 2 and more generally M(n)(0) = E(), n ≥ 1 (8) The mgf uniquely determines a distribution in that no two distributions can have the same mgf So knowing a mgf characterizes the distribution in question@ µ ¬ ·#' @ µ c Q b µ S V #' @ ì6ë < \ %T í È Y í !ñ v "á ì6ë @ W Z K r 8 r M r S #' @4#&ì 0b G \ v Q b4#&ì b3õ p Ç Ý Ò v ~ M G \ v A r O X r ~ #' @ ì6ë æ



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(e) the variance of Y 4 Let Y be a random variable having mean µ and suppose that E(Y −µ)4 ≤ 2 Use this information to determine a good upper bound to P(Y −µ ≥ 10) 5 Let U and V be independent random variables, each uniformly distributed on 0,1 Set X = U V and Y = U − V Determine whether or not X and Y are® µ Ø Å ¯ w ¥ é Y w g µ Ø Å P 6 b ;2 X is a vector of independent random variables iff V is diagonal (ie all offdiagonal entries are zero so that sij =0 for i 6= j) Proof From (1), if the X0s are independent then sij =Cov(Xi;Xj)=0 for all i 6= j, so that V is diagonal



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Harnett, ch 3) A The expected value of a random variable is the arithmetic mean of that variable,B u \ { u u O s O Q Õ ê Á µ á v Ü Ú b q O d } ( Î v y è ¾ ® ð v ^ F õ v d ^ s z Ù v Q C r d } o r è ¾ W ?(b) (7 points) Derive , the variance of U, in terms of b, and the covariance 2 σU 2, 2 σX σY σXY 2 σU = EU 2 − E(U)2 = EU2 because the second term is zero = E(Y − µ Y) 2 − 2b(X−µ x)(Y − µ Y) b 2(X−µ x) 2 = E(Y − µ Y) 2 − 2bE(X−µ x)(Y − µ Y) b 2E(X−µ x) 2 = 2 − 2b σY σXY b 2 2 σX (c) (6 points) Suppose I want to choose b in order to



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