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↳. Independent trials, each resulting in a success with probability p, are per- formed until k consecutive successful trials have occurred. Let X be the total number of successes in these trial, and let P₁ = P(X = n). (a) Find Pk. (b) Derive a recursive equation for the Pñ, n ≥ k, by imagining that the trials continue forever and conditioning on the time of the first failure. Verify your answer in part (a) by solving the recursion for Pk. (d) When p= .6, k = 3, find Pg. (c)

↳. Independent trials, each resulting in a success with probability p, are per- formed until k consecutive successful trials have occurred. Let X be the total number of successes in these trial, and let P₁ = P(X = n). (a) Find Pk. (b) Derive a recursive equation for the Pñ, n ≥ k, by imagining that the trials continue forever and conditioning on the time of the first failure. Verify your answer in part (a) by solving the recursion for Pk. (d) When p= .6, k = 3, find Pg. (c)

MATLAB: An Introduction with Applications
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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54. Independent trials, each resulting in a success with probability p, are per- formed until k consecutive successful trials have occurred. Let X be the total number of successes in these trial, and let Pñ = P(X = n). n (a) Find Pk. (b) Derive a recursive equation for the PÅ, n ≥k, by imagining that the trials continue forever and conditioning on the time of the first failure. (c) Verify your answer in part (a) by solving the recursion for Pk. (d) When p= .6, k = 3, find Pg.
Transcribed Image Text:54. Independent trials, each resulting in a success with probability p, are per- formed until k consecutive successful trials have occurred. Let X be the total number of successes in these trial, and let Pñ = P(X = n). n (a) Find Pk. (b) Derive a recursive equation for the PÅ, n ≥k, by imagining that the trials continue forever and conditioning on the time of the first failure. (c) Verify your answer in part (a) by solving the recursion for Pk. (d) When p= .6, k = 3, find Pg.
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