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Suppose that the random variables Y Y, satisfy where xi, ,Xn are fixed constants and Ei,...
4. (24 marks) Suppose that the random variables Yi,..., Yn satisfy Y-B BX,+ Ei, 1-1, , n, where βο and βι are parameters, X1, ,X, are con- stants, and e1,... ,en are independent and identically distributed ran- dom variables with Ei ~ N (0,02), where σ2 is a third unknown pa- rameter. This is the familiar form for a simple linear regression model, where the parameters A, β, and σ2 explain the relationship between a dependent (or response) variable Y...
Suppose Xi, X2, . . . , xn are i.id. random variables with Xi ~「α, β). Find the distribution of the sum of the X,'s and the distribution of the average of the X,'s.
3. Let Xi, . . . , Xn be iid randoln variables with mean μ and variance σ2. Let, X denote the sample mean and V-Σ, (X,-X)2. (a) Derive the expected values of X and V. (b) Further suppose that Xi,-.,X, are normally distributed. Let Anxn ((a)) an orthogonal matrix whose first rOw 1S be , ..*) and iet Y = AX, where Y (Yİ, ,%), ard X-(XI, , X.), are (column) vectors. (It is not necessary to know aij...
Let Xi, X2, , xn be independent Normal(μ, σ*) random variables. Let Yn = n Ση1Xi denote a sequence of random variables (a) Find E(%) and Var(%) for all n in terms of μ and σ2. (b) Find the PDF for Yn for all n c) Find the MGF for Y for all n
QUESTION 2 Let Xi.. Xn be a random sample from a N (μ, σ 2) distribution, and let S2 and Š-n--S2 be two estimators of σ2. Given: E (S2) σ 2 and V (S2) - ya-X)2 n-l -σ (a) Determine: E S2): (l) V (S2); and (il) MSE (S) (b) Which of s2 and S2 has a larger mean square error? (c) Suppose thatnis an estimator of e based on a random sample of size n. Another equivalent definition of...
Let Xi, ..., Xn be random variables with the same mean and with covariance function where |ρ| < 1 . Find the mean and variance of Sn-Xi + . . . + Xn. Assume thatE(X. ) μ and V(X) σ2 for i (1.2. , n}
Problem 3.1 Suppose that XI, X2,... Xn is a random sample of size n is to be taken from a Bermoulli distribution for which the value of the parameter θ is unknown, and the prior distribution of θ is a Beta(α,β) distribution. Represent the mean of this prior distribution as μο=α/(α+p). The posterior distribution of θ is Beta =e+ ΣΧ, β.-β+n-ΣΧ.) a) Show that the mean of the posterior distribution is a weighted average of the form where yn and...
Exercise 7.7 Of the variables (yi, xi) only the pair (yi, xi) are observed. In this case, we say that yi is a latent variable. Suppose where ui is a measurement error satisfying Let ß denote the OLS coefficient from the regression of yi on (a) Ís β the coefficient from the linear projection of yi on z? (b) Is β consistent for β as n oo? (c as n oo. e) Find the asymptotic distribution of yn(3-8 as
3. Let X1, . . . , Xn be iid random variables with mean μ and variance σ2. Let X denote the sample mean and V-Σ,(X,-X)2 a) Derive the expected values of X and V b) Further suppose that Xi,...,Xn are normally distributed. Let Anxn - ((a) be an orthogonal matrix whose first row is (mVm Y = (y, . . . ,%), and X = (Xi, , Xn), are (column) vectors. (It is not necessary to know aij for...
Question 4 [15 marks] The random variables X1,... , Xn are independent and identically distributed with probability function Px (1 -px)1 1-2 -{ 0,1 fx (x) ; otherwise, 0 while the random variables Yı,...,Yn are independent and identically dis- tributed with probability function = { p¥ (1 - py) y 0,1,2 ; otherwise fy (y) 0 where px and py are between 0 and 1 (a) Show that the MLEs of px and py are Xi, n PY 2n (b)...