8 Consider the time series model Y1 ~ N(0, 102) and YlYt-1, ,y ~ N øt_i,σ2)...
Consider the regression model where the εi are i.i.d. N(0,σ2) random variables, for i = 1, 2, . . . , n. (a) (4 points) Show βˆ is normally distributed with mean β and variance σ2 . 1 1SXX Question 6 Consider the regression model y = Bo + B12 + 8 where the €, are i.i.d. N(0,0%) random variables, for i = 1,2, ..., n. (a) (4 points) Show B1 is normally distributed with mean B1 and variances
1. Consider the linear regression model iid 220 with є, 면 N(0, σ2), i = 1, . . . , n. Let Yh = β0+ßX, be the MLE of the mean at covariate value Xh . (f) Suppose we estimate ơ2 by 82-SSE/(n-2). Derive the distribution for You can use the fact that SSE/σ2 ~ X2-2 without proof. (g) What is a (1-a)100% confidence interval for y? (h) Suppose we observe a new observation Ynet at covariate value X =...
Consider the simplified Bayesian model for normal data The joint posterior pdf is ful, σ21 x)a(σ2,-/2-1 expl_jy.tx, _aPI The marginal posterior pdfs of μ and σ 2 can be obtained by integrating out the other variable (8.30) y@1 x) α (σ2)-m;,-1/2 expl-- Σ.-tri-x)2 (8.31) d. Let q1 and q2 be they/2 and 1-y/2 quantiles of (8.31). Show that the 1-γ credible interval (gi,q2) is identical to the classic confidence interval (5.19) (with ar replaced by y). Hence, a (1-α) stochastic...
If X ~ N(0, σ2), then Y function of Y is X follows a half-normal distribution; i.e., the probability density This population level model might arise, for example, if X measures some type of zero-mean difference (e.g., predicted outcome from actual outcome) and we are interested in absolute differences. Suppose that Yi, ½, ,y, is an iid sample from fy(ylơ2) (a) Derive the uniformly most powerful (UMP) level α test of 2 2 0 versus Identify all critical values associated...
σ2). 6. Suppose X1, Yİ, X2, Y2, , Xn, Y, are independent rv's with Xi and Y both N(μ, All parameters μί, 1-1, ,n, and σ2 are unknown. For example, Xi and Yi muay be repeated measurements on a laboratory specimen from the ith individual, with μί representing the amount of some antigen in the specimen; the measuring instrument is inaccurate, with normally distributed errors with constant variability. Let Z, X/V2. (a) Consider the estimate σ2- (b) Show that the...
1. Consider a GLM (generalised linear model) for a Poisson random sample Y1,. .. , Y, with \Vi each Yi having a pdf or pmf f(y; A;) = i= 1, . .. ,n. Yi = 0, 1,2, -..; ^; > 0; Y;! Note that the pdf from an exponential family has the following general form b(0) + c(y, a(o) y0 exp f(y; 0, 6) = Suppose the linear predictor of the GLM is n = a+Bxi, with (a,B) being the...
This is equation 8: full question, which contains y1 ya(t) = n(e) / 102 . Use Equation (8) above or go through the reductio-of-order process to find a second solution 72 of the preceding equation such that {/1,} is a fundamental set of solutions of y" - (1+ y + 4y = 0 on (0,0). Y2 = Y = Solve the initial-value problem y"(t) + 4y'(t) + 13y(t) =0, y(0) = 3, y'(O) = 6. Express you answer in the...
Let y,p ~iid Exp (0), for i = 1, . . . , n. (p(y|0) for 6 to be Gamma(a, b), tha distribution of θ BeAy). Assume the prior distribution Find the posterior 2. t is, p(0) -ba/ra)ge-i exp{-be. 3. Find the posterior predictive distribution of a future observation in problem 2
Exercise5 Consider a linear model with n -2m in which yi Bo Pi^i +ei,i-1,...,m, and Here €1, ,En are 1.1.d. from N(0,ơ), β-(A ,A, β), and σ2 are unknown parameters, zı, known constants with x1 +... + Xm-Tm+1 + +xn0 , zn are 1, write the model in vector form as Y = Xß+ε describing the entries in the matrix X. 2, Determine the least squares estimator β of β. Exercise5 Consider a linear model with n -2m in which...
Question 3 [25] , Yn denote a random sample of size n from a Let Y, Y2, population with an exponential distribution whose density is given by y > 0 if o, otherwise -E70 cumulative distribution function f(y) L ..,Y} denotes the smallest order statistics, show that Y1) = min{Y1, =nYa) 3.1 show that = nY1) is an unbiased estimator for 0. /12/ /13/ 3.2 find the mean square error for MSE(e). 2 f-llays Iat-k)-at 1-P Question 4[25] 4.1 Distinguish...