One way to remember Markov’s Inequality (also called Chebyshev’s Inequality) is to remember this application: No more than 1/5 of the population can have more than 5 times the average income. For instance, if the average income of a certain country is USD $3000 per month, no more than 20% of the citizens can earn more than $15 000!

**Brief Explanation**

is Markov’s Inequality, where is the probability measure. Taking to be 5 times the average income, the left hand side represents the probability of having more than 5 times the average income. The right hand side is .

**Chebyshev’s/Markov’s Inequality (Proof):**

If is a measure space, is a non-negative measurable extended real-valued function, and , then

Proof:

Define

Then . Thus . Dividing both sides by gives the result.