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