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Y u vg. 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 2 Proof Was given in lectures (and a different proof can be found in Notes 4) Some additional properties of conditional expectations 1 If X and Y are independent rv’s then E(XjY)=E(X) Proof. U À ð ì v } µ v ~ À µ } v v ( Z v P X 15t Z v Á o o Ç } µ Z À À } v v u Ç o } o o v P µ P M d Z WD K< ' µ t ^ À v Z } v U Á Z Z v o µ d Z ^ v ( } W } i D v P u v U Á o o µ o Z v v P o Z v translated into the following languages Arab. µ v À * # vº ´ ¿ M z ½ Y zº k Y a y Zº ¸ ¯ zº v · Y ºº z ¾ ´ d ^ ¯ ½ y Z d z { º ¯ M z À g P c ¾ y z v · Z ¶ « Z ¯ ² ¿ Y.

Y µ } v E } X & } u W } ( } u } v Y µ } v µ } v l } v µ À & } o o } Á µ } Y µ } v · } v ( v o z lE í D õ E l h v W } i o À o ô Ç } µ Z À o ï 9 } ( Z E/ ( µ v v P µ P v Z v W. 3DJH GH 6HUYLFHV G¶LQYHVWLVVHPHQW 7' ,QF W } o µ µ o Æ u v o v o P o u v o P. µ ′ n = EX n The nth central moment of X is defined as Let X have cdf FX(x), Y = g(X)and let domain and codomain of g(X), respectively, be X = {x fX(x) > 0}, and Y = {y y = g(x) for some x ∈ X} (a) If g is an increasing function on X then FY (y) = FX g−1(y) for y ∈ Y (b) If g is a decreasing function on X, then FY (y) = 1−FX g−1(y) for y ∈ Y Proof The cdf of Y = g(X.

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`d @&8M3T A'VB h@ ro3 A'*'08 ¸p76 @B)o¹. 5 r r G y w r G r { v G s x { z { > r r y r { 5 r v v x r 5 r w x { r > r s { > 0 î î ò s d í ï î ì î ì r í r > ì í r< î î ò rs d r ì õ ð ô ì í v P } v o U } ( } v o v v v µ o. Then, U = g(X) and V = h(Y) are also independent for any function g and h We will come back to various properties of functions of random variables at the end of this chapter 2 2 Moments and Conditional Expectation Using expectation, we can define the moments and other special functions of a random variable Definition 2 Let X and Y be random variables with their expectations µ X =.

Y @z ¶ ‰g y;. I =1,,n is a normal random sample then ¯ x ∼ N (µ,σ 2 /n) Let µ =µ 1,µ 2,,µ n = E (x) be the expected value of x =x 1,x 2,,x n and let Σ=σ ij;. Title Microsoft Word FINAL G Things to Know PFL Author t Created Date 1/12/21 PM.

‰ µ @v z @t v x @v z @x v y @v z @y v z @v z @z ¶ = ¡ @p @z „ µ @2v z @x 2 @2v z @y @2v z @z2 ¶ ‰g z With v y = v z = 0 (from assumption 2), @v x=@x = 0 from the simplifled continuity equation, (E611), g y = ¡g, g x = g z = 0 (assumption 4), and steady °ow (assumption 1), these momentum balances simplify enormously, to „ @2v x @y2 =. è ¿ Å µ » ¶ ¤ è ¶ Í µ × ¿ ­ · q Z ~ è µ × ¿ ­ å × ¶ Í µ × ¿ ­ · q å D Ô ¢ T £ Ô ¢ D £ 3 ¿ ¢ 3 ¿ w ÿ µ æ Æ H s ¶ ¶ \ µ × ¿ ­ ¬ V G q A L 7 P S U F Y G í y è ¿ Å µ » ¶ 075 3 ¿ ô Ï y y y y ² y 7 4 q y ³ y y. A 9 A P g V s Y 4 ¥ * J ¿ ­ J k µ · ÿ ^ V { V ¥ Q * × = ® ^ V { V ¥ 1(& b 4 g k x ý ~ Q , T * ö p ' ' A * j k * J k a ) V ' ­ I ² ô!.

í ñ ì ì u h í ñ' > } v P µ u h í ñ î í ñ ì ì u h í ót ^ Z } h í ñ' lh í ót ï µ h í ï' ð ï X ì ì u í ñ ì u h í í' í ñ ì u h í í ï X ï ì u > } v P µ u h í í ñ ^ Z } h í óD í ï X î ì u î ì ì u h í ñ' µ h í ï î î ì ì u h í ót î ì ì u h í ñ î ì ì u h í óD ð X í ñ u > } v P µ u h í óD ï ^ Z } h í í �. Title Microsoft Word FAQ_Iss03 Author pgodof Created Date 4// PM. Z µ } v v Y^ ^ D Z /Z h> K í ï u u E í ì u u hDWZ d u } µ u µ o } o } ~ } í î r ó ñ o } Y A ì U ô ô G í U õ ì lt Z l î ð Z hDWZ d u } µ u µ o } o } ~ } r ï ì o } Y A ì U ò ï G í U ð ó lt Z l î ð Z hDWZ Y µ.

Title Microsoft Word Revaluation17FAQpdf Author rbell Created Date 3/3/17 AM. U µ = 5 v x y r G y w r G r { v G s x { z { 0 r r v > r E r y r { 5 r v v x r 5 r w x { r > r s { > 0 î î ò r s d Erasmus Title RESULTS SITE K26 VET xls. Ì Ì U } µ o U ( µ v l U P } } À U Z r Z } Y Y µ v Ç } v v À Ç } µ } Æ o } Á } o } ( u µ Z } µ P Z Z v r u.

Y)= a i E (x i)= µ a i and V (y)= a 2 i V (x i)= σ 2 a 2 i Any linear function of a set of normally distributed variables is normally distributed If x i ∼ N (µ,σ 2);. Word unscrambler will unscramble letters and find possible words Enter your scrambled letters and uncover the hidden words!. Title Microsoft Word ΠΠΠΤΠΠΠΠΠΩΣΠΣΧΠΠΠΥΠΠΠ_v2 Author children Created Date.

H , T * ð 7 k µ · ÿ G 2 , T * Ý ô J g V 4 * Ì % 5 ö ð Z "t ¢ ò Q · " % ¹ ¨ Q å ¨ * J ' * G " % , T * J ¿ ­ J k µ £ ÿ Ó $ * ý ß % % ' Q ï Ó * A Ò % = ® ^ V { V ¥ 7 I L > Q â % , T '1(. Title Microsoft Word FAQ_Iss01 Author pgodof Created Date 4/10/ PM. ^ Z ( îZ o t , î ì î ì r^ îZh r î ì î ì r/ v µ u v rW Ì rd v/ r î ì î í Y µ } v v Á.

& "/ ) &7& '2)& /&00& ) /%,* &#,)& * / 1% /"1/,#)"5 * /,0& *"),0&0 / ,/" *"),0&0 /"3&"!2 2) 1 *"),0&0 ,/!. 1 random v ec tor with mean µ y and v ar iance co v a. %( 365 /^µ>V& ^Y @a=^5&ª{«aª XZ ?8 /eVB@B4µ>VB'Ca8oT=?7?.

µ o o v r u o W } 8 u Ç v Á o ( X } P X µ l ^ µ v Ç U í ì µ v î ì í ô. @µ @y @r @y @µ fl fl fl fl fl fl @v @y @t @z @u @z @v @z @t fl fl fl fl fl fl fl fl fl fl fl fl fl fl (2349) This determinant is called the Jacobian of the transformation of coordinates Example 1 The Jacobian of cylindrical coordinates The relation between Cartesian and cylindrical coordinates was given in (2305) We can easily compute the Jacobian, J = fl fl. Y = g(X), where X is asymptotically normally distributed X » AN ¡ „;¾2 ¢ This is bookwork If the derivative, _g, of g is nonzero, then Y » AN(g(„);fg_(„)g2 ¾2) We saw this result in a slightly difierent form in the lectures, stated as p n( ¡„) ¡!L N(0;¾2) =) p n(g()¡g(„)) ¡!L N(0;fg_(„)g2¾2) and these two results are equivalent 4 MARKS (b) Suppose that X1.

∂v ∂y, Πyz = µ ∂v ∂z ∂w ∂y , (15) Πzz = −pλ∇·v2µ ∂w ∂z, Πxz = µ ∂u ∂z ∂w ∂x (16) µ and λ are the coefficients of dynamic and bulk viscosity respectively These expressions assume that the relationship between stress and velocity gradients is • linear (which is valid for Newtonian fluids) and • isotropic (ie, the intrinsic properties of the. D Z & u ( } µ Z µ î ì î ì µ o v o µ r Æ v P } v r o Æ v } ( ¨ î ï í u o o } v ~ ¨ í ô õ u o o } v ( r Æ } v Z t o Z D v P u v µ v P u v X. And µvalues The vapour resistance of a material is a measure of the material’s reluctance to let water vapour pass through The vapour resistance takes into account the material’s thickness, so can only be quoted for a particular thickness of material It is usually measured in MNs/g (“Mega Newton seconds per gram”) If the quantity is measured in MNs/gm (notice the small “m” at.

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 independent 6 Let U and V be independent random variables, each uniformly distributed on 0,1 Determine the mean and variance of the random variable Y. ^ } v î w y µ o Ç } v } o & µ v u v o •/ v } µ } v } y 3/$1 '2 $&7 &(& '2 whvw srwhqwldo vroxwlrqv fuhdwh surfhvv vwuxfwxuh hvwdeolvk v\vwhpv frqgxfw wudlqlqj phdvxuh txdolw\ fkdudfwhulvwlfv froohfw gdwd &(& prqlwru dqg dqdo\h gdwd vwxg\ wkh uhvxowv gudz frqfoxvlrqv 3/$1 lghqwli\ sureohpv ghilqh ghvluhg rxwfrphv lghqwli\ srwhqwldo vroxwlrqv. Title Microsoft Word Resolução SE 71 Modo Revisão (2) Author vitor Created Date 10/13/ PM.

Find V (X c )Sho w y our w o rk T hi s is imp orta n t b ecause it tells us w e can a lw a y s pr etend the mea n eq uals ze ro when calculat ing co v aria nce ma trices 6Let X b e a p !. Title Microsoft Word Questions and answers 1 bsgee 16 02 Author dimmerth Created Date 10/28/16 PM. Where g called link function and µ(X) = IE(YX) 15/52 Oneparameter canonical exponential family Canonical exponential family for k = 1, y ∈ IR (yθ −b(θ) ) f θ (y) = exp c(y,φ) φ for some known functions b(·) and c(·,·) If φ is known, this is a oneparameter exponential family with θ being the canonical parameter If φ is unknown, this may/may not be a twoparameter.

~v = ~y ⇔ vµ = yµ ⇔ vα = y α, ~v = A~y ⇔ vµ = n ∑ ν=1 Aµνyν ⇔ vν = n ∑ µ=1 Aνµyµ (16) This index notation is also applicable to other manipulations, for instance the inner product Take two vectors~v and ~w, then we define the inner product as ~v· ~w = v 1w 1 ···v nw = n ∑ µ=1 v µw (17) (We will return extensively to the inner product Here it is just. Title Microsoft PowerPoint Freed60Final19pptx Author gmoore Created Date 1//19 PM. G711 is a narrowband audio codec originally designed for use in telephony that provides tollquality audio at 64 kbit/s G711 passes audio signals in the range of 300–3400 Hz and samples them at the rate of 8,000 samples per second, with the tolerance on that rate of 50 parts per million (ppm) Nonuniform (logarithmic) quantization with 8 bits is used to represent each sample, resulting.

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V o Z µ v v Á } l } u v v o Ç } u Z u v } v Z ( } M î õ µ P µ î ì í ó ï í µ P µ î ì í ó ï í µ P µ î ì í ó E. Title Microsoft Word Quiz No 9 Author JayKay Created Date 7/15/ PM.