Show that o0 cos T
Show that o0 cos T
16. Using the contour in Figure 14.22 show that o0-
16. Using the contour in Figure 14.22 show that o0-
Let x ~ Nk(0, Σ) with pdf f(x) where Σ = {Σ defined as . The entropy h(x) is h(x) =-J f(x) In f(x) In(2me)"E! (a) Show that h(x) ( b) Hence, or otherwise, show that |E| s 11k! Σί, with equality holding if and only if Σ¡j 0, for i j [Hadamard's inequality]
(4) Let(an}n=o be a sequence in C. Define R-i-lim suplanlì/n. Recall that R e [0,x] o0 is the radius of convergence of the power series Σ a (z 20)" Assume that R > 0 (a) Prove that if 0 < ρ < R, then the power series converges uniformly on the closed (b) Prove that the power series converges uniformly on any compact subset of the disk Ix - xo< R
(4) Let(an}n=o be a sequence in C. Define R-i-lim...
2. Suppose Xi ~ N(8,02) where θ > 0. (a) Show that s--(x, Σ¡! xi) is a sufficient statistic of θ where X is the sample mean. (b) Is S minimal sufficient? (c) Can you find a non-constant function g(.) such that g(S) is an ancillary statistic?
(20 points) Let Z be a standard normal random variable and X -ZI(Z). Find E(X) (a, o0)
(20 points) Let Z be a standard normal random variable and X -ZI(Z). Find E(X) (a, o0)
Question 1. (exercise 26 in textbook) Let A be a σ algebra of subsets of Ω and let B E A Show that F = {An B : A e A} is a σ algebra of subsets of B Is it still true when B is a subset of Ω that does not belong to A?
Question 1. (exercise 26 in textbook) Let A be a σ algebra of subsets of Ω and let B E A Show that F = {An B : A e A} is a σ algebra of subsets of B Is it still true when B is a subset of Ω that does not belong to A?
b) Using the binomial theorem show that Σ (-1)"/2 (n) cos" k(z) sink(z), Σ (-1)(k-1)/2C) cox"-"(x) sink(z). cos(nx) = sin(nx) = COS k-odd 6 marks]
(a) Show that Σ-1(n-X)2 İs a biased estimator of σ2 (b) Find the amount of bias in the estimator. (c) What happens to the bias as the sample size n increases?