You have a random variable X whose expectation you\'d like to estimate. You have

Question

You have a random variable X whose expectation you'd like to estimate. You have the ability to draw samples from X's distribution, and you know nothing about E(X), and you only know that Var(X) 10. (You may assume Var(X) = 10, since this is the worst case.) To estimate E(X), you take n i.i.d. samples of X and average them. Using the bound based on Chebyshev's Inequality:

a. Suppose you take 1000 samples. How condent are you that your estimate is

within an absolute error of 0.5?

b. Suppose instead that you want an absolute error of at most 2 and a condence

parameter of 0.02 (you want to be "98% confident"). How many samples do you need?

c. Suppose instead that you take 2500 samples and you want a condence parameter of 0.1 ("90% confident"). What absolute error bound will you get with this confidence?

Explanation / Answer

Given Var(X) = 10

s.d = sqrt(Var) = sqrt(10) = 3.162

a.) n = 1000

error = 0.5

E = z*s.d/sqrt(n)

0.5 = z*(3.162)/sqrt(1000)

So z = 5

So 100% confident

b.) error = 2

98% CI, z=2.3263

E = z*s.d/sqrt(n)

2 = 2.3263*3.162/sqrt(n)

So n = 13.53 = 14 (rounded)

c.) 90% CI, z = 1.645

n = 2500

E = z*s.d/sqrt(n) = 1.645*3.162/sqrt(2500) = 0.104

Related Questions

Navigate

• Browse (All)
• Browse Y
• Subjects

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.