Answered: 23. Let Y denote a geometric random…

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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23. Let Y denote a geometric random variable with probability of success p. (a) Show that for a positive integer a, P(Y > a) = (1 − p)a (b) Showthatforapositiveintegeraandb,P(Y >a+b|Y >a)=(1−p)a =P(Y >b). (c) Why do you think the property in (b) is called the memoryless property of the geometric distribution? (d) In the development of the distribution of the geometric random variable, we assumed that the experiment consisted of conducting identical and independent trials until the first success was observed. In light of these assumptions, why is the result in part (b) “obvious”? (e) Show that P (Y = an odd integer) = p 1−q2 24. Given that the random variable W is binomial distribution with n trials and success probability p in each trial and P (W = w) = h(w), show that (a) h(w) =p(n−w+1),w>0 h(w − 1) (1 − p)w (b) E [W(W − 1)]= n(n − 1)p2. ? 1 ? 1−(1−p)n+1 (c)E W+1 = (n+1)p

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