Statistics

The Markov’s Inequality states that for a value a>0, we have for any random variable X that takes no negative values, the following upper bound is always observed:

                  - Pr⁡(X≥a) ≤ E(X)/a

Markov’s inequality says that for a positive random variable X and any positive real number a, the probability that X is greater than or equal to a is less than or equal to the expected value of X divided by a. Markov’s inequality is very important to estimate probabilities, considering its generality in the sense that it applies to any non-negative random variable X.

Illustration of the Inequality

To illustrate the inequality, suppose we have a distribution with nonnegative values (such as a chi-square distribution). If this random variable X has expected value of 3 we will look at probabilities for a few values of a.

- For a = 10 Markov’s inequality says that P (X ≥ 10) ≤ 3/10 = 30%. 
   So there is a 30% probability that X is greater than 10.
- For a = 30 Markov’s inequality says that P (X ≥ 30) ≤ 3/30 = 10%. 
   So there is a 10% probability that X is greater than 30.
- For a = 3 Markov’s inequality says that P (X ≥ 3) ≤ 3/3 = 1. 
Events with a probability of 1 = 100% are certain. 
So this says that some value of the random variable is greater than or equal to 3. 
This should not be too surprising. If all the values of X were less than 3, then the expected value would also be less than 3.