4 5 of 5Question 44. (a) Assume T > 0, and that the moment generating function Mx (t) of a random variable X is finite for t < T. Explainthe expansionMx (t) = 1+ ut +s?t² + o(t³),2%3|where u =E(X) and s2E(X²). [You may assume the validity of interchanging expectation and differentiation.](b) Let X, Y be independent, identically distributed random variables with mean 0 and variance 1, and assume theirmoment generating function M satisfies the condition of part (a) with T = o. Suppose that X +Y and X - Y areindependent.(i) Using M(2t) deduce that (t) := M(t)/M(-t) satisfies (t) = »(t/2)². Clearly state any property of themoment generating functions you use.(ii) Show that b(h)= 1+ o(h?) as h → 0, and deduce that (t)= 1 for all t.(iii) Show that X and Y are normally distributed.国

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Description: 4 5 of 5Question 44. (a) Assume T > 0, and that the moment generating function Mx (t) of a random variable X is finite for t < T. Explainthe expansionMx (t) = 1+ ut +s?t² + o(t³),2%3|where u =E(X) and s2E(X²). [You may assume the validity of intercha

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1. (a) Assume T > 0, and that the moment generating function Mx (t) of a random variable X is finite for t < T. Explain he expansion Mx(t) = 1+ µt + + o(t*),

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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