Is there a way to lower bound the left tail probability of a random variable?

Question

I am looking for a bound of the form $P(X<0) > t$ where $X$ is a general random variable with positive mean, and all of whose (or most) moments exist. $t$ is ideally a function of these moments. I am looking for a non-trivial bound of this kind.

Answer 1

If X has negative mean, we would have by Chebyshev:

P(X $\geq$ 0) $\leq$ P($|X-EX| \geq |EX|$) $\leq$ $\frac{E(X-EX)^2}{(EX)^2}$

(Thus, $P(X < 0) > 1 - \frac{E(X-EX)^2}{(EX)^2}$)

Other bounds involving higher moments should also be possible (and I would guess that whichever of your moments imposes the tightest conditions would "win"), but this is a simple and yet non-trivial bound)