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Random variables, expectation, and variance DSE 210 Random variables Roll a die Define X = ⇢ 1ifdieis 3 0 otherwise Here the sample space is⌦= {1, 2, 3, 4, 5, 6}.

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H ßL!5–9% U z%¦. As represented in Figure 11 for µ. E(x−µ)2 = varx = σ2 (17) Now we expand the square on the lefthand side giving Ex2−2µExµ2 = σ2 Making use of (149) then gives (150) as required Finally, (151) follows directly from (149) and (150) Ex2−Ex2 = µ2 σ2 −µ2 = σ2 19 For the univariate case, we simply differentiate (146) with respect to xto.

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KE X / E X ' KW X í. And σ2, the first and second order moments, respectively, obtainable from the pdf as µ. U ~ r ~ í.

^/ } v t µ. Those who have taken economics courses may remember the equation above, which lists the components of GDP GDP (Y) is the combination of consumption (C), investment (I), government spending (G), and net exports (exports (X) less imports (M)). ~ 9 ¬!8/¡C_ E­.

View Consider a random sample X1docx from MATHS 104 at Harvard University Consider a random sample X1,X2,, from F(θ) &. II Let x1, x2, , x n be a random sample drawn from a population with mean µ. K 4 c x U ¥.

The moment generating function of X is M(t)= ˆ 1 t =0 et−1 t t 6= 0 The characteristic function of X is φ(t)= ˆ 1 t =0 eit−1 it t 6= 0 The population mean, variance, skewness and kurtosis of X are. View MUESTRALESxlsx from ESTADISTIC 12 at Grancolombiano Polytechnic hombres µ= s= n= 93 14 43 108 Para mujeres Valor esperado de la media = E(x) = µ. ^ E d Z v P .

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T@ Eurynorhynchus pygmeus ßL!. O , o o >>/ Ed E Z'z KZWKZ d ^ Zs/ ^ U /E X >dD µ. And variance σ2In other words, E(xi) = µ, and Var (xi) = σ 2 for i = 1, 2, , n, and the x’s are all independent of each otherLet ∑ n i xi n x 1 1 be the sample mean (a) (4 points) Show that E(x) = µE( x ) = E (∑n i xi n 1 1) = n 1 E(∑) = n i xi 1 n 1 ∑ n i E xi.

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X N L b v { g . ’ () * ’ , /) * ’ 0 1 2 3 4 5 6 7 8 9;. = 2 and σ 2= 15 The Gaussian pdf N(µ,σ2)is completely characterized by the two parameters µ.

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C _ X @ p Z b g ̏ i y W ł B z Z ^ ʔ DCM I C ւ悤 I IDCM z } b N ADCM J } ADCM _ C L ADCM T ADCM 낪 ˂ Ńz Z ^ P ʂ̂c b l z f B O X ^ c ̃l b g ʔ̂ł B. E(X Y) = E(X)E(Y) (4) If X and Y are independent, then Var(X Y) = Var(X)Var(Y) (5) The above properties generalize in the obvious fashion to to any finite number of rvs In general (independent or not) Var(X Y) = Var(X)V(Y)2Cov(X,Y), where Cov(X,Y) def= E(XY)−E(X)E(Y), is called the covariance between X and Y, and is usually. P } yy r , µ.

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D ` b v E n E K X E H ̌ p r p ` b v \\ E ͂ ݁E E E ē n E d H E E H ̌ ɡ ɒ d ` b v ē E Z ~ b N ̓ ނ̌ ɡ ̃ X ɔ פ X s h E ͂ɂ ėD Ă 褕 L p r Ɏg p ł ܂ ؕ t ^ C v x#300 n ގ Y f j b P d d グ Ѓg b v } ̃_ C n X DCM I C ł͔̔ Ă ܂ B ̑ H 戵 Ă ܂ B 270mm 48mm s25mm d 60g. And variance σ2 If we say X ∼ N(µ, σ2) we mean that X is distributed N(µ. P } } E X ì.

T _ ¤. 6 3 RANDOM VECTORS Expectation of a Quadratic Form Theorem LetEX = µandcov(X) = ΣandAbeaconstant matrix Then E(X−µ)0 First Proof (brute force). ^ / E DKdKZ/ D d D d/ /E&KZD d/ Z >/'/KE l dd X >d ZE X /^ 'EK ^dKZ X >>.

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C e r t i f i c a t i o n D a t e 2 0 1 8 0 8 0 9 L a t e s t Is s u e 2 0 1 8 0 8 0 9 E x p i r y D a t e 2 0 2 1 0 8 0 8 Pa g e 2 o f 2 T h i s c. } v ñ. The shorthand X ∼inverse Gaussian(λ,µ)is used to indicate that the random variable X has the inverse Gaussian distribution with parameters λand µ.

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W } v W } v ^ } W ( } v } ( ^ l o o r W } d Z v µ. ’ % () (* , / (0 1 2 3 % 4 (5 0 2 (6 2 3 1 7 * 0 8 9 (;. σ πσ µσ • The notation N(µ, σ2) means normally distributed with mean µ.

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