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3. Let X Exponential(A) and let t be a constant with 0 < t < A. Let b > 0 be any value. (a) Calculate P(X >b) exactly. (b) Now suppose we didn't know the answer to (a), i.e., we could not calculate P(X > b) exactly. Recalling that E(X)= }, use Markov's inequality to establish an upper bound on P(X > b). (c) Next we'll apply Markov's inequality in a different way. First, calculate E(etX). %3D

3. Let X Exponential(A) and let t be a constant with 0 < t < A. Let b > 0 be any value. (a) Calculate P(X >b) exactly. (b) Now suppose we didn't know the answer to (a), i.e., we could not calculate P(X > b) exactly. Recalling that E(X)= }, use Markov's inequality to establish an upper bound on P(X > b). (c) Next we'll apply Markov's inequality in a different way. First, calculate E(etX). %3D

A First Course in Probability (10th Edition)
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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3. Let X ~ Exponential() and let t be a constant with 0 <t < A. Let b > 0 be any value. (a) Calculate P(X > b) exactly. (b) Now suppose we didn't know the answer to (a), i.e., we could not calculate P(X > b) exactly. Recalling that E(X)= , use Markov's inequality to establish an upper bound on P(X > b). (c) Next we'll apply Markov's inequality in a different way. First, calculate E(etx). (d) Next, use the Markov inequality to prove a bound on P(etX > a) (here a > 0 is any positive number, while we assume 0 <t < A as before). 1 (e) Finally, using your answer to part (d), derive a bound on P(X > b) (here b > 0 is any positive number, and again 0 <t <X). You will have to choose the value of a to use. (f) Now let's compare. Pick some values for A, b, and t. Compute the exact value of P(X > b) as in part (a), the upper bound on P(X > b) as in part (b), and the alternative upper bound on P(X > b) as in part (e). How close are these upper bounds to the real probability? Which bound is better? Try this for a few values of X, b,t and describe what you see. 6.
Transcribed Image Text:3. Let X ~ Exponential() and let t be a constant with 0 <t < A. Let b > 0 be any value. (a) Calculate P(X > b) exactly. (b) Now suppose we didn't know the answer to (a), i.e., we could not calculate P(X > b) exactly. Recalling that E(X)= , use Markov's inequality to establish an upper bound on P(X > b). (c) Next we'll apply Markov's inequality in a different way. First, calculate E(etx). (d) Next, use the Markov inequality to prove a bound on P(etX > a) (here a > 0 is any positive number, while we assume 0 <t < A as before). 1 (e) Finally, using your answer to part (d), derive a bound on P(X > b) (here b > 0 is any positive number, and again 0 <t <X). You will have to choose the value of a to use. (f) Now let's compare. Pick some values for A, b, and t. Compute the exact value of P(X > b) as in part (a), the upper bound on P(X > b) as in part (b), and the alternative upper bound on P(X > b) as in part (e). How close are these upper bounds to the real probability? Which bound is better? Try this for a few values of X, b,t and describe what you see. 6.
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