I'm taking the Algorithms: Design and Analysis II class, one of the questions asks:
Suppose $ X $ is a random variable that has expected value $ 1 $.
a) What is the probability that $ X $ is $ 2 $ or larger? (Choose the strongest statement that is guaranteed to be true.)
- At most $ 100\% $
- At most $ 25\% $
- $ 0 $
- At most $ 50\% $
b) Does the answer change if $ X $ is always nonnegative?
This is not an active homework question. I've completed the course successfully, but I don't like loose ends, hence going back and trying to solve the problems that I couldn't answer at that time.
For part $b$, using Markov's Inequality we have $$P(X\geq 2)\leq \frac{E[X]}{2}=\frac{1}{2}$$
For part $a$, consider the discrete random variable defined by $$P(X=2)=.9$$ $$P(X=-8)=.1$$ Then $E(X)=1$ yet $P(X\geq 2)=.9$. This counterexample eliminates the second, third, and fourth options. So it must be the (trivial) first option: "at most $100\%"$