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Assume we want to test whether the die is balanced Then the value of the Pearson 2 statistic

INSTRUCTIONS TO CANDIDATES
ANSWER ALL QUESTIONS
Consider the following linear regression model y X3 -- e.
where y E R” is an n-dimensional response vector, X E R” XP is an n x p- dimensional matrix (p n) for explanatory variables. e N(O. aI) is the random vector, and W is a known matrix.
It is known that the \ILE of ß is 3 = (XX) ‘Xy, the distribution of y ‘— N(X(3. 0.21)  and the residual vector is  = — X(XXF’X]y Suppose that rank(X) = p such that the  inverse of XX exists. Let z = Uy.
1- Show that z and (3 are independent if and only if UX O.
2- Assume UX = O. Show that if U is an n x n projection matrix then j IzII2
follows the X2-distñbutiofl with tr(U) degrees of freedom.
Fill in the following blanks. Please find the answer of the problems and fill it in the blanks.
(a) Suppose X1 N(3.4). X2 N(2.6) and X3 - N(1,7). Let Y = X1 + X2 — 3X3. Then. E(Y) = , V(Y) = , )‘ -‘ , and P(Y > 5)
(b) Suppose we flipped a die 1000 times, and the counts of 1, 2. 3. 4. 5 and 6 are 160.
170. 155. 163, 180, 172. Assume we want to test whether the die is balanced. Then, the value of the Pearson 2 statistic is . Under ¡lo it  approximately foLlows
___________ distribution. Based on this data, we ___________ ho.
(c) Assume ri,» . ‘ux are observations from the Poisson distribution with mean 0. Sup pose we have r = 3.5. Assume we want to test hl : 0 = 3 versus Il : 0  3. Then,the value of —2 log A is . Under I1, the test statistic approximately follows ______________ distribution. Based on this data, we ______________ ib.
(d) Suppose X1, ,, are ud with common PI)F fe(r) = O/(l +x) for r >0. where 0 > 0 is an unknown parameter. The sufficient statistic is T = . the  MEE isÔ = . and the Fisher information is I(0)= . let 0 be the true parameter. The approximate distribution of O is ,/(0 — Oo).
(e) Suppose X1, ,, are lid with common Pl)F f.(x), where f(r) > 0 for all r > 0 and fe(x) = O for r < 0. 1.et O be the MEE and 0 be the true parameter. Assume 1(0) = 02.
Then, /‘(Õ—00) . and Æt—0) g . lfweohservedÒ=4, then the 95% confidence interval for O is approximately
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