Strengthen Markov’s inequality with conditional probability

Markov’s inequality is very general and hence very weak. Assume that X is a non-negative random variable, a > 0, and X has a finite expected value, Then Markov’s inequality says that

\text{P}(X > a) \leq \frac{\text{E}(X)}{a}

In [1] the author gives two refinements of Markov’s inequality which he calls Hansel and Gretel.

Hansel says

\text{P}(X > a) \leq \frac{\text{E}(X)}{a + \text{E}(X - a \mid X > a)}

and Gretel says

\text{P}(X > a) \leq \frac{\text{E}(X) - \text{E}(X \mid X \leq a)}{a - \text{E}(X \mid X \leq a)}

Related posts

[1] Joel E. Cohen. Markov’s Inequality and Chebyshev’s Inequality for Tail Probabilities: A Sharper Image. The American Statistician, Vol. 69, No. 1 (Feb 2015), pp. 5-7