Question

Hoeffding versus Chebyshev 1/4 points (graded) Let X_1, X_2, ..., X_n ? 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 (|X_n - E [X]| ? c * ? / ?n) where ?^2 = VarX_i for c = 2 and c = 6. Each answer is numerical. Using Chebyshev inequality: P (|X_n - E [X]| ? 2 * ? / ?n) ? 0.25 P (|X_n - E [X]| ? 6 * ? / ?n) ? 0.1667 Using Hoeffding inequality: P (|X_n - E [X]| ? 2 * ? / ?n) ? P (|X_n - E [X]| ? 6 * ? / ?n) ?

          Hoeffding versus Chebyshev
1/4 points (graded)
Let X_1, X_2, ..., X_n ? 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 (|X_n - E [X]| ? c * ? / ?n) where ?^2 = VarX_i
for c = 2 and c = 6. Each answer is numerical.
Using Chebyshev inequality:
P (|X_n - E [X]| ? 2 * ? / ?n) ? 0.25
P (|X_n - E [X]| ? 6 * ? / ?n) ? 0.1667
Using Hoeffding inequality:
P (|X_n - E [X]| ? 2 * ? / ?n) ?
P (|X_n - E [X]| ? 6 * ? / ?n) ?

Hoeffding versus Chebyshev
1/4 points (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) ? 0.25
P (|Xn - E [X]| ? 6 * ? / ?n) ? 0.1667
Using Hoeffding inequality:
P (|Xn - E [X]| ? 2 * ? / ?n) ?
P (|Xn - E [X]| ? 6 * ? / ?n) ?

Added by Cheryl C.

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