I am struggling with part (ii) Let g(x, y) (e" +1)2+2(e-e(e1). 22-1 For any fixed x...
·J (I) < 0 for all such y. (Hint: let g(x)--f(x) and use part (a)) 3. In this problem, we prove the Intermedinte Value Theorem. Let Intermediate Value Theorem. Let f : [a → R be continuous, and suppose f(a) < 0 and f(b) >0. Define S = {t E [a, b] : f(z) < 0 for allェE [a,t)) (a) Prove that s is nonempty and bounded above. Deduce that c= sup S exists, and that astst (b) Use Problem...
2. Let f(x,y) = e-r-u, 0 < x < oo, 0 < y < oo, zero elsewhere, be the pdf of X and Y. Then if Z = X + Y, compute (a) P(Z 0). (b) P(Z 6) (c) P(Z 2) (d) What is the pdf of Z?
(e) Let x = (T1,T2, . . . ,xn),y=(y1,y2, . . . ,Un) ER" (i) Show that for any λ E R: 3 where llxll = 1/(x, x). x, y (ii) Use (7) for λ =- to show: 1a1 with equality, if and only if, there exists a λ E R such that y = 1x.
2) Let X and Y be independent exponential random variables with means E[X] = 0 and EY = 28. 1 1 f(310) = -X/0 e x > 0, f(y|0) = e-4/20 y > 0 0 24 a) Show that the likelihood function can be written as (2 points) L(0) = e-3(x+3) 202 b) Find the MLE ô of 0. (5 points)
5. Let y E C2([0, T]; R), T > 0 satisfy y"(t) = 피t, y(0) = y'(0) = 0 e R. Use Picard-Lindelöf 1+t' to prove that a unique solution to the IVP exists for short time, as follows: (a) Let b E R2, A E M2 (R) . Show that any function g : R2 -R2.9(x) = Ax+b is Lipschitz. 1 mark (b) Transform the DE for y into a(t) Az(t) +b(t) for a suitable z, A, b. 2...
I am struggling with problem 1. I have solved for Part a but cannot figure out how to solve for Part b. Any help is appreciated. HW 4. Due Nov 6th. 1. Let X have density f(x) = { kx(1-r) OSISI otherwise (a) Find k. (b) Find E[X] (c) Find Var(X) (4) Find F(x) and define this for all r in (-20%) (e) Find the density of Y = log X.
* if <<1 Let h(x) = { 2 – 22 if i< x < 2 2 - 3 if < > 2 Use the limit definition of derivative to find h'(1) if it exists.
A. Let (X, d) be a metric space so that for every E X and every r>0 the closed ball N,(z) = {ye X : d(y, z) < r} is com pact. Let be a homeomorphism. (1) Prove that f"-+m-fn。fm for all n, m E Z. (2) Let z E X and suppose that F, {fn (z) : n E 2) is a closed subset of X Prove that F is a discrete subset of X (A subset Y C...
Let f, g E H(C) be such that |f(z)| < \g(z)| for any z e C. Show that there exists a E D(0,1) such that f(z) = ag(z) for any z E C. (Hint: consider f/g and be careful with the zeros of g.)
Q1 (7 points) For k e R any constant, find the general solution to xa y" + (1 – k)x y' = 0, and use it to show that when k < 0, all solutions tend to a constant as x + 2O.