How to use Chebyshev's inequality to find a lower bound?

Question

I have that $\text{var}(X) = \frac13v^2$ and I want to find the lower bound for $P(|X| \le v)$

I tried doing the following: $P(|X| \le v) = 1 - P(|X| \ge v)$ so using Chebyshev's inequality I have that

$1 - P(|X| \ge v) \le \frac13$

$\frac23 \le P(|X| \ge v)$,

but I don't know what can I say about $P(|X| \le v)$ since everything is now on terms of $P(|X| \ge v)$.

Answer 1

You are making a mistake in using Chebyshev's inequality. What you get $P(|X| \geq v) \leq 1/3$ (assuming that the mean is $0$) and not $1-P(|X| \geq v) \leq 1/3$