Markov inequality for random variables with negative values.

Question

I'm given the maximum value of a random variable $X$ (for example $50$) and its mean, $\mathbb E(X)=20$. How do I find the upper bound to $P(X\le -10)$?

Answer 1

Define new RV: S = 50 - R

E(S) = 30

P(R<=-10) = P(S>=60)

P(S>=60) = 30/60 = 1/2

You commented: I needed to clarify one last thing. P(50-X >= 60) <= 1/3

It should be 1/2 not 1/3.

Answer 2

Hint:

• $50-X$ is a nonnegative random variable since $50$ is an upperbound.

• Express your inequality in the form of $Pr(50-X \ge c)$.