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The Chebyshev Inequality applies to any continuous random variable X, whatever its distribution, and states that for any given constant k ≥ 1 the probability that X is more than k standard deviations away from its mean is no more than 1/k². P(|X - μ| ≥ ko) ≤ 1 k² = 1.2. (a) Suppose X is a random variable with uniform distribution from 0 to 2, and k Calculate the value on each side of the inequation above, and confirm that the Chebyshev Inequality holds in this case. (b) Do the same for a random variable Y~ Exp(0.5) with k = 1.2. (c) Check the inequality, with the same k, for a standard normal random variable Z. (d) For which of X, Y, and Z does the Chebyshev Inequality have the smallest gap which one is nearest to breaking it?

The Chebyshev Inequality applies to any continuous random variable X, whatever its distribution, and states that for any given constant k ≥ 1 the probability that X is more than k standard deviations away from its mean is no more than 1/k². P(|X - μ| ≥ ko) ≤ 1 k² = 1.2. (a) Suppose X is a random variable with uniform distribution from 0 to 2, and k Calculate the value on each side of the inequation above, and confirm that the Chebyshev Inequality holds in this case. (b) Do the same for a random variable Y~ Exp(0.5) with k = 1.2. (c) Check the inequality, with the same k, for a standard normal random variable Z. (d) For which of X, Y, and Z does the Chebyshev Inequality have the smallest gap which one is nearest to breaking it?

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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The Chebyshev Inequality applies to any continuous random variable X, whatever its distribution, and states that for any given constant k ≥ 1 the probability that X is more than k standard deviations away from its mean is no more than 1/k². P(|X − µ| ≥ ko) ≤ 1 k² = (a) Suppose X is a random variable with uniform distribution from 0 to 2, and k Calculate the value on each side of the inequation above, and confirm that the Chebyshev Inequality holds in this case. (b) Do the same for a random variable Y~ Exp(0.5) with k = 1.2. (c) Check the inequality, with the same k, for a standard normal random variable Z. (d) For which of X, Y, and Z does the Chebyshev Inequality have the smallest gap which one is nearest to breaking it? 1.2.
Transcribed Image Text:The Chebyshev Inequality applies to any continuous random variable X, whatever its distribution, and states that for any given constant k ≥ 1 the probability that X is more than k standard deviations away from its mean is no more than 1/k². P(|X − µ| ≥ ko) ≤ 1 k² = (a) Suppose X is a random variable with uniform distribution from 0 to 2, and k Calculate the value on each side of the inequation above, and confirm that the Chebyshev Inequality holds in this case. (b) Do the same for a random variable Y~ Exp(0.5) with k = 1.2. (c) Check the inequality, with the same k, for a standard normal random variable Z. (d) For which of X, Y, and Z does the Chebyshev Inequality have the smallest gap which one is nearest to breaking it? 1.2.
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