6. (a) If f(x a) for -al R, show that i-0 fO(a) fori 0,1,2, (where f0 (a) f(a), and fori 1 f(a) denotes the i-th derivative of f at a). (b) If f e* , find f(2014) 6. (a) If f(x a) for -al R, show that i-0 fO(a) fori 0,1,2, (where f0 (a) f(a), and fori 1 f(a) denotes the i-th derivative of f at a). (b) If f e* , find f(2014)
0 Let (f.) be a group, show that (ly) G where Gly); = Lael | ag= ga & gely is the center of G. (So, show that cly)< ; & cgjat. ) @ let y be a group, gel & Haf. Prove that Ks4 where us Ki Cig) := {acly I ag = gay is the centralizer of g inily, and K: N (H): = hatly I aH=Hay in the normalizen of Henly.
Problem i. C Y-A + β2xitei, İ-1, , n, where ei's are i i d N(0, σ2). Let D A + β2T, where βί and A 2 are the LSEs of;, and β2, respectively. Show that the random variables B2 and D are uncorrelated, and explain why B2 and D must therefore be independent. onsider the simple linear regression model, ï.c.
Question B 7. (a) Let -1 0 0 (i) Find a unitary matrix U such that M-UDU where D is a diagonal matrix. 10 marks] (i) Compute the Frobenius norm of M, i.e., where (A, B) = trace(B·A). [4 marks] 3 marks] (iii) What is NM-illp? (b) Let H be an n × n complex matrix (6) What does it mean to say that H is positive semidefinite. (il) Show that H is positive semidefinite and Hermitian if and only...
In the simple linear regression with zero-constant item for (xi , yi) where i = 1, 2, · · · , n, Yi = βxi + i where {i} n i=1 are i.i.d. N(0, σ2 ). (a) Derive the normal equation that the LS estimator, βˆ, satisfies. (b) Show that the LS estimator of β is given by βˆ = Pn i=1 P xiYi n i=1 x 2 i . (c) Show that E(βˆ) = β, V ar(βˆ) = σ...
5. (6 points) Find all values of 0, where 0 es 2n, where tan 8-1. Show your work.
For s > 0 define the gamma function I (s) by T () = [co-dt. Show that I (8) extends to an analytic function in the half-plane 20 = {ZEC: Rez >0}, and that the above formula continues to hold there. Hint: Show that S T. (s) ds = 0 for every triangle T in C where I (8) = le-+48-1dt for S E C and 0 <€ < 1.
10. Let Ae-at if t > 0 h(t) = 10 ift < 0 where A and a are parameters as in Example 2.21. Show that Α h(x) = (Ch)(i) = Tarla+id)
19.2. Let f : [a,b] → R be integrable. Show that rb 72 (r)dz, un 0O i=1 where a, b > 0 and h (b/a)1/n. In particular, calculate J-3r2dr by consid- ering a partition P which divides the interval [2, 3] into n parts in geometric progression at the points 2, 2h, 2h2,2h3,... ,2h"-1,2h" -3 19.2. Let f : [a,b] → R be integrable. Show that rb 72 (r)dz, un 0O i=1 where a, b > 0 and h (b/a)1/n....
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