EXERCISE 6.1.1 3 Let p, q E X and γ : [0, 1]- X a path from p to q. a. For a loop α in X based at...
Thee part question. Please answer all parts! Let E be a field of characteristic p > 0 (we proved p must always be prime). Verify that the ring homomorphism X : Z → E determined by sending χ : 1-1 E (the unity in E) ( so x(n)-n 1E wheren1E 1E 1E (n-times), x(-n)- nle for any n 1,2,3,... and X(0) 0E by definition of χ) is in fact a ring homomorphism with ker(X) = pZ. Úse the fundamental homomorphism...
Let 0 < γ < α . Then a 100(1 − α )% CI for μ when n is large is Xbar+/-zγ*(s/sqrt(n))The choice γ = α /2 yields the usual interval derived in Section 8.2; if γ ≠ α /2, this confidence interval is not symmetric about . The width of the interval is W=s(zγ+ zα-γ)/sqrt(n). Show that w is minimized for the choice γ = α /2, so that the symmetric interval is the shortest. [ Hints : (a)...
Let X1, . . . , Xn be a sample taken from the Gamma distribution Γ(2, θ−1) with pdf f(x,θ)= θ^2xexp(−θx) if x ≥ 0, θ ∈ (0,∞), and 0 otherwise, (A) Show that Y = ∑ni=1 Xi is a complete and sufficient statistic. (B) Find E(1/Y) . Hint: If W ∼ χ2(k) then E(W^m) = 2mΓ(k/2+m) for m > −k/2. Note also that Y Γ(k/2) Γ(n) = (n − 1)!, n ∈ N∗ . Facts from 1(C) are useful:...
Exercise 8 The pdf of Gamma(α, λ) is f(x)-ra)r"-le-Az for x 0. a. Let X ~ Gamma (a, λ). Show that E( )--A for α > 1 b. Let Ux2. Show that E()for n > 2 n-2
1.(c) 2.(a),(b) 5. Let Xi,..., X, be iid N(e, 1). (a) Show that X is a complete sufficient statistic. (b) Show that the UMVUE of θ 2 is X2-1/n x"-'e-x/θ , x > 0.0 > 0 6. Let Xi, ,Xn be i.i.d. gamma(α,6) where α > l is known. ( f(x) Γ(α)θα (a) Show that Σ X, is complete and sufficient for θ (b) Find ElI/X] (c) Find the UMVUE of 1/0 -e λ , X > 0 2) (x...
let a > 0 and define g(x) := x^(a+1) - (a+1)x + a. Use the mean value theorem to show that g(x)>0 for all x>0, where x~=1 3. Let α > 0 and define g(x):-Χα +1-(α + 1)x + α. Use the mean value theorem to show that (x>for allx >0, where x I 3. Let α > 0 and define g(x):-Χα +1-(α + 1)x + α. Use the mean value theorem to show that (x>for allx >0, where x...
1. Let Q be the set of polynomials with rational coefficients. You may assume that this is an abelian group under addition. Consider the function Ql] Q[x] given by p(px)) = p'(x), where we are taking the derivative. Show that is a group homomorphism. Determine the kernel of 2. Let G and H be groups. Show that (G x H)/G is isomorphic to H. Hint: consider defining a surjective homomorphism p : Gx HH with kernel G. Then apply the...
Part D,E,F,G 10. Let p(x) +1. Let E be the splitting field for p(x) over Q. a. Find the resolvent cubic R(z). b. Prove that R(x) is irreducible over Q. c. Prove that (E:Q) 12 or 24. d. Prove: Gal(E/Q) A4 or S4 e. If p(x) (2+ az+ b)(a2 + cr + d), verify the calculations on page 100 which show that a2 is a root of the cubic polynomial r(x)3-4. 1. f. Prove: r(x) -4z 1 is irreducible in...
Problem 4.9 (e) /(z) = and γ is parametrized by r(t), 0 z + t 1, and satisfies Imr(t)> 0, r(0) -4 + i, and γ(1) 6 + 2i (f) f(s) sin(z) and γ is some piecewise smooth path from 1 to π. 4.2 and the fact that the length of γ does not change under 4.9. Prove Proposi reparametrization. (Hint: Assume γ, σ, and τ are smooth. Start with the definition off, f, apply the chain rule to σ...
Let α and β be positive constants. Consider a continuous-time Markov chain X(t) with state space S = {0, 1, 2} and jump rates q(i,i+1) = β for0≤i≤1 q(j,j−1) = α for1≤j≤2. Find the stationary probability distribution π = (π0, π1, π2) for this chain.