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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 \( X \sim \) Exponential(\(\lambda\)) and let \( t \) be a constant with \( 0 < t < \lambda \). Let \( b > 0 \) be any value. (a) Calculate \(\mathbb{P}(X > b)\) exactly. (b) Now suppose we didn’t know the answer to (a), i.e., we could not calculate \(\mathbb{P}(X > b)\) exactly. Recalling that \(\mathbb{E}(X) = \frac{1}{\lambda}\), use Markov’s inequality to establish an upper bound on \(\mathbb{P}(X > b)\). (c) Next we’ll apply Markov’s inequality in a different way. First, calculate \(\mathbb{E}(e^{tX})\). (d) Next, use the Markov inequality to prove a bound on \(\mathbb{P}(e^{tX} \ge a)\) (here \( a > 0 \) is any positive number, while we assume \( 0 < t < \lambda \) as before). (e) Finally, using your answer to part (d), derive a bound on \(\mathbb{P}(X \ge b)\) (here \( b > 0 \) is any positive number, and again \( 0 < t < \lambda \)). You will have to choose the value of \( a \) to use. (f) Now let’s compare. Pick some values for \(\lambda\), \(b\), and \(t\). Compute the exact value of \(\mathbb{P}(X > b)\) as in part (a), the upper bound on \(\mathbb{P}(X > b)\) as in part (b), and the alternative upper bound on \(\mathbb{P}(X > b)\) 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 \(\lambda\), \(b\), \(t\) and describe what you see.
Transcribed Image Text:3. Let \( X \sim \) Exponential(\(\lambda\)) and let \( t \) be a constant with \( 0 < t < \lambda \). Let \( b > 0 \) be any value. (a) Calculate \(\mathbb{P}(X > b)\) exactly. (b) Now suppose we didn’t know the answer to (a), i.e., we could not calculate \(\mathbb{P}(X > b)\) exactly. Recalling that \(\mathbb{E}(X) = \frac{1}{\lambda}\), use Markov’s inequality to establish an upper bound on \(\mathbb{P}(X > b)\). (c) Next we’ll apply Markov’s inequality in a different way. First, calculate \(\mathbb{E}(e^{tX})\). (d) Next, use the Markov inequality to prove a bound on \(\mathbb{P}(e^{tX} \ge a)\) (here \( a > 0 \) is any positive number, while we assume \( 0 < t < \lambda \) as before). (e) Finally, using your answer to part (d), derive a bound on \(\mathbb{P}(X \ge b)\) (here \( b > 0 \) is any positive number, and again \( 0 < t < \lambda \)). You will have to choose the value of \( a \) to use. (f) Now let’s compare. Pick some values for \(\lambda\), \(b\), and \(t\). Compute the exact value of \(\mathbb{P}(X > b)\) as in part (a), the upper bound on \(\mathbb{P}(X > b)\) as in part (b), and the alternative upper bound on \(\mathbb{P}(X > b)\) 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 \(\lambda\), \(b\), \(t\) and describe what you see.
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