For upper bound of probability $P\{X \ge t\}$, we have Markov's inequality, Chernoff's bound, and moment bound.
But how can we deal with lower bound? Is there any similar inequalities for lower bound analysis?
For upper bound of probability $P\{X \ge t\}$, we have Markov's inequality, Chernoff's bound, and moment bound.
But how can we deal with lower bound? Is there any similar inequalities for lower bound analysis?
Since the indicator function $I_{x \ge t} \ge 1 - (x-a)^2/(t-a)^2$ for $a > t$, $P(X \ge t) \ge 1 - \frac{\sigma^2+(a-\mu)^2}{(t-a)^2}$ If $\mu > t$, the optimal $a = \mu + \sigma^2/(\mu-t)$, obtaining $P(X \ge t) \ge \frac{(\mu-t)^2}{(\mu-t)^2+\sigma^2} \ \text{for}\ t < \mu$