HW4 sol.pdf - Homework 4 Math 561: Theory of Probability I...

Homework 4
Math 561: Theory of Probability I
Due date: February 25, 2021
Each problem is worth 10 points and only five randomly chosen problems will be graded if there are more than
5 problems. Please indicate whom you worked with, it will not affect your grade in any way.
1. (i) Prove that Markov’s inequality P ( X > t ) 6 E ( X ) /t is sharp for fixed t > 0, i.e., there is a non-negative
random variable X such that P ( X > t ) = E ( X ) /t .
(ii) (One-sided Chebyshev inequality) Suppose that E ( X ) = 0, Var( X ) = σ 2 < ∞ and a > 0. Prove that
P ( X > a ) 6
σ
2
σ 2 + a 2
and there is a r.v. X for which equality holds.