You have the random variable X whose expectation like to evaluate. You have skill to draw samples from X's distribution, and you know nothing about E(X), and you know that Var(X) 10. (You may suppose Var(X) = 10, as this is the worst case.) To evaluate E(X), you take n i.i.d. samples of X and average them. Using bound based on Chebyshev's Inequality:
a. Assume you take 1000 samples. How confident are you that the estimate is within the absolute error of 0.5?
b. Assume instead that you wish the absolute error of at most 2 and the confidence parameter of 0.02 (you wish to be "98% confident"). How many samples do you require?
c. Assume instead that you take 2500 samples and you wish the confidence parameter of 0.1 ("90% confident"). Find absolute error bound will get with the confidence?