Exercise I. Let X, Y ~ N(0, 1) be jointly Gaussian random variables and let t€ R. Thegoal of this exercise is to prove the following identity, using three different ways:E[Xe] =tE[XY] E[e¹]. (*)1. a. Prove that E[XeX] = tE[e¹X].b. Set p = E[XY]. Show that (X, Y) Law (X, pX+√√1 - p²Z), where Z~ N(0, 1)is independent from X. Deduce that (*) holds true.2. We recall from the lecture notes that ety=eH(y) for all y € R. Byapplying this identity to y = Y, give a second proof of (*).3. For s R, compute E[esX+tY] using the known expression for the Laplace transformof a Gaussian random variable, and then give a third proof of (*) by differentiatingwith respect to s.

Given::-Let X, Y~ N(0, 1) be jointly Gaussian random variables and let t€ R.

Answered: Exercise I. Let X,Y~ N(0, 1) be jointly…

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

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13 Let X and Y be jointly continuous random variables with joint Paj. fxy (x, y ) = x² + for 0sxs1, o sys2 3 eisewhere (a) Check X and Y for independence (b) are f (x/y) and f (y/x) are valid Pdfs.
1
Let X and Y be random variables with joint den- √(x+2y) x,y>0,x+y Y). (b) Calculate fx y(xly). sity: f(x,y): = (c) Calculate P(X < 1 Y = 1). (d) Calculate E[X²|Y].
A fair coin is tossed three times. The random variable X is defined to be , where h is the number of flips that come up heads. For example, X(HHT) = 2 . What is E[X]?
Exercise 17. Let X₁,..., Xn~ N(u, o2) be i.i.d. where u is known, but o2 is unknown. Consider the estimate n 1 Σ(X₁-1 for the variance. a) Show that 7² C := i=1 -μ)² 72 2 ·Σ( X ₁ = μ)² ~ x² (n). O i=1 b) For a € (0, 1), denote the a-quantile of x2 (n) by qn(a). Using this notation, find numbers an, bn E R such that P(o² bno²) = 2.5%. c) Using these results, develop a test which can be used to test the hypothesis Ho: o² = o against the alternative H₁: 02 08. d) Assume we have n = 100, μ = 0 and we have observed 2 = 4.81. Test the hypothesis Ho: o² = 4 vs. H₁: o² #4 at significance level 5%.
Let a, µ E R. Calculate f exp{-ax²/2+µx}dx. (Hint: recall that the PDF for a normal random variable Z ~ N(H, o2) is p(2) = (270²)-1/2 exp{-(z – µ)²/(20²)} and that p(z)dz = 1.)
3-1.4 Let X be the outcome from rolling one die and Y the outcome from rolling a second die. a) Find the joint probability of the event X 3. b) Find E[XY]. c) Find E[).
3. Let the random variable X have the pdf f(x) = 2(1 — x), 0 ≤ x ≤ 1, zero elsewhere. a) Find the cdf of X. Provide F(x) for all real numbers x (set up the appropriate cases). b) Find P(1/4 < X < 3/4). c) Find P(X= 3/4). d) Find P(X ≥ 3/4).
b) X-XN (0,1), and W₁ σχ YHYN(0,1), for i=1,2,3,...,10, then: Let Z₁ = i) State, with parameter(s), the probability distribution of the statistic, T = ay 10 ΣW2 √Σt,w₁² ii) Find the mean and variance of the statistic T = 10 Σ{12/2 iii) Calculate the probability that a statistic T = Z₁ + W₁ is at most 4. iv) Find the value of ß such that P(T> B) = 0.01, where T = E₁Z²+₁ W².
Assume that S={O1, O2, O3, O4} and that w1 = 0.22, w2 = 0.12, w3 = 0.17, w4 = 0.49. Let E={O2, O3} and F={O3, O4}. (a) What is the probability Pr[E]? (b) What is Pr[F′]? (c) What is Pr[(E∪F)′]?
J 2
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