er the SLR model, Y₁, Y2, ... Yn are independent normal rando , E[Y;] = ß0 + ß₁xi and V(Y₁) = σ², for i = 1, 2, ..., n. Compute the likelihood function, L(y1, Y2, ..., Yn|ẞ0, B1, σ²) = L(ßo; Show that the MLE of σ² is SSE. [the MLEs of n Bo and B₁ are the same as the least-squares estimators, Bo and B₁.
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- Let Y1, . . . , YN be a random sample from the Normal distribution Yi ∼ N(ln β, s2) where s2 is known. Find the Score function, the estimating equation and the information matrix.Let Y1, . . . , YN be a random sample from the Normal distribution Yi ∼ N(ln β, s2) where s2is known. Find the maximum likelihood estimator of b from first principles.Find the Score function, the estimating equation and the information matrix.Consider the one-way analysis of variance model Xij = µ + a; + Eij, i= 1,.., m, j= 1,..., ni, where ɛij ~ N(0,0²) are independent. Let n= n1 + · ·+ nm, ... ni 1 ni 1 X;. >Xii for i = 1,..., m, and X. = - ΣΣΧ. ni j=1 i=1 j=1 (a) Show that SS(TO) = SS(T) + SS(E), where SS(TO) = E E i=12j=1 (Xij – X..)², m m ni SS(T) Σι (X.-Χ.) and SS(E) -ΣΣ (Χ- X.) . i=1 i=1 j=1
- Let X, Y be independent random variables with exponential distribution of parameter θ > 0. Are the random variables Z = X + Y and W = X / (X+Y) independent?If X, Y are standardized random variables and r(aX+bY,bX+aY)= 1+2ab a²+b² I Find r(X,Y), coefficient of correlation between X and Y.Let X1 and X2 be two independent normal random variables with parameters (0,1) and (0,4) respectively. Let Y, = 2X, + X, and Y, = X, - 3X,. Find %3D
- 5. Let Y1, . . . , YN be a random sample from the Normal distribution Yi ∼ N(ln β, s2) where s2is known.Find the maximum likelihood estimator of b from first principles.Find the Score function, the estimating equation and the information matrix.Exercise 39 Let X1 ~ N(µ1,07) and X2 ~ N(u2, o3). Find the expected value and variance of Y = X1+ X2.Consider two independent exponential random variables X1 and X2 with parameter lambda=1. LetY1 = X1 Y2 = X1 + X2. Find the MMSE estimate of Y1 using Y2.
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