A computer program crashes at the end of each hour with probability p if it has not crashed already. Let H be the number of hours until the first crash. (a) What is the distribution of H? Derive E[H] and Var[H] in terms of p. (b) Use Chebyshev's Theorem to upper-bound Pr(|H-1/p| ≥ x/p) for x > 0. Use your bound to show that Pr(H 2 a/p) ≤ 1-p/ (a-1)^2 (c) Compute the exact value of Pr(H 2 a/p). For simplicity, assume a is a constant multiple of p. Compare your result with the bound in part (b).
A computer program crashes at the end of each hour with probability p if it has not crashed already. Let H be the number of hours until the first crash. (a) What is the distribution of H? Derive E[H] and Var[H] in terms of p. (b) Use Chebyshev's Theorem to upper-bound Pr(|H-1/p| ≥ x/p) for x > 0. Use your bound to show that Pr(H 2 a/p) ≤ 1-p/ (a-1)^2 (c) Compute the exact value of Pr(H 2 a/p). For simplicity, assume a is a constant multiple of p. Compare your result with the bound in part (b).
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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![A computer program crashes at the end of each hour with probability p if it has not crashed already. Let H be the number
of hours until the first crash.
(a) What is the distribution of H? Derive E[H] and Var[H] in terms of p.
(b) Use Chebyshev's Theorem to upper-bound Pr(|H-1/p| 2 x/p) for x > 0.
Use your bound to show that Pr(H ≥ a/p) ≤ 1-p/(a−1)^2
(c) Compute the exact value of Pr(H 2 a/p). For simplicity, assume a is a constant multiple of p. Compare your result with
the bound in part (b).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7b239036-d094-4540-921c-d6c41f94772a%2F2a139d98-f8b7-4d3b-a424-1ec3477a4903%2Fpoesteq_processed.jpeg&w=3840&q=75)
Transcribed Image Text:A computer program crashes at the end of each hour with probability p if it has not crashed already. Let H be the number
of hours until the first crash.
(a) What is the distribution of H? Derive E[H] and Var[H] in terms of p.
(b) Use Chebyshev's Theorem to upper-bound Pr(|H-1/p| 2 x/p) for x > 0.
Use your bound to show that Pr(H ≥ a/p) ≤ 1-p/(a−1)^2
(c) Compute the exact value of Pr(H 2 a/p). For simplicity, assume a is a constant multiple of p. Compare your result with
the bound in part (b).
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