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

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

Hansel says

and Gretel says

## 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