Question

Markov Inequality For a random variable X ? 0 with mean ? > 0, and any number t > 0: P(X ? t) ? ?/t. Note that the Markov inequality is restricted to non-negative random variables. Chebyshev inequality For a random variable X with (finite) mean ? and variance ?^2, and for any number t ? 0, P(|X - ?| ? t) ? ?^2/t^2. Remark: When Markov inequality is applied to (X - ?)^2, we obtain Chebyshev's inequality. Markov inequality is also used in the proof of Hoeffding's inequality. Hoeffding versus Chebyshev 4 points possible (graded) Let X1, X2, ..., Xn iid? Unif(0, b) be n i.i.d. uniform random variables on the interval [0, b] for some positive b. Suppose n is small (i.e. n < 30) so that the central limit theorem is not justified. Find an upper bound on the probability that the sample mean is "far away" from the expectation (the true mean) of X. More specifically, find the respective upper bounds given by the Chebyshev and Hoeffding inequalities on the following probability: P(|?Xn - E[X]| ? c?/?n) where ?^2 = VarXi for c = 2 and c = 6. Each answer is numerical. Using Chebyshev inequality: P(|?Xn - E[X]| ? 2?/?n) ? P(|?Xn - E[X]| ? 6?/?n) ? Using Hoeffding inequality: P(|?Xn - E[X]| ? 2?/?n) ? P(|?Xn - E[X]| ? 6?/?n) ?

          Markov Inequality
For a random variable X ? 0 with mean ? > 0, and any number t > 0:
P(X ? t) ? ?/t.
Note that the Markov inequality is restricted to non-negative random variables.
Chebyshev inequality
For a random variable X with (finite) mean ? and variance ?^2, and for any number t ? 0,
P(|X - ?| ? t) ? ?^2/t^2.
Remark:
When Markov inequality is applied to (X - ?)^2, we obtain Chebyshev's inequality. Markov inequality is also used in the proof of Hoeffding's inequality.
Hoeffding versus Chebyshev
4 points possible (graded)
Let X1, X2, ..., Xn iid? Unif(0, b) be n i.i.d. uniform random variables on the interval [0, b] for some positive b. Suppose n is small (i.e. n < 30) so that the central limit theorem is not justified.
Find an upper bound on the probability that the sample mean is "far away" from the expectation (the true mean) of X. More specifically, find the respective upper bounds given by the Chebyshev and Hoeffding inequalities on the following probability:
P(|?Xn - E[X]| ? c?/?n) where ?^2 = VarXi
for c = 2 and c = 6. Each answer is numerical.
Using Chebyshev inequality:
P(|?Xn - E[X]| ? 2?/?n) ?
P(|?Xn - E[X]| ? 6?/?n) ?
Using Hoeffding inequality:
P(|?Xn - E[X]| ? 2?/?n) ?
P(|?Xn - E[X]| ? 6?/?n) ?

Markov Inequality
For a random variable X ? 0 with mean ? > 0, and any number t > 0:
P(X ? t) ? ?/t.
Note that the Markov inequality is restricted to non-negative random variables.
Chebyshev inequality
For a random variable X with (finite) mean ? and variance ?^2, and for any number t ? 0,
P(|X - ?| ? t) ? ?^2/t^2.
Remark:
When Markov inequality is applied to (X - ?)^2, we obtain Chebyshev's inequality. Markov inequality is also used in the proof of Hoeffding's inequality.
Hoeffding versus Chebyshev
4 points possible (graded)
Let X1, X2, ..., Xn iid? Unif(0, b) be n i.i.d. uniform random variables on the interval [0, b] for some positive b. Suppose n is small (i.e. n < 30) so that the central limit theorem is not justified.
Find an upper bound on the probability that the sample mean is "far away" from the expectation (the true mean) of X. More specifically, find the respective upper bounds given by the Chebyshev and Hoeffding inequalities on the following probability:
P(|?Xn - E[X]| ? c?/?n) where ?^2 = VarXi
for c = 2 and c = 6. Each answer is numerical.
Using Chebyshev inequality:
P(|?Xn - E[X]| ? 2?/?n) ?
P(|?Xn - E[X]| ? 6?/?n) ?
Using Hoeffding inequality:
P(|?Xn - E[X]| ? 2?/?n) ?
P(|?Xn - E[X]| ? 6?/?n) ?

Added by Joshua H.

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