Markov’s Inequality: No more than 1/5 of the population can have more than 5 times the average income

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

\mu(\{x\in X: f(x)\geq\epsilon\})\leq\frac{1}{\epsilon}\int_X f\,d\mu is Markov’s Inequality, where \mu is the probability measure. Taking \epsilon=5A 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 \frac{1}{5A}\cdot A=\frac 15.

Chebyshev’s/Markov’s Inequality (Proof):
If (X,\Sigma,\mu) is a measure space, f is a non-negative measurable extended real-valued function, and \epsilon>0, then \displaystyle \mu(\{x\in X: f(x)\geq\epsilon\})\leq\frac{1}{\epsilon}\int_X f\,d\mu.

Proof:
Define \displaystyle s(x)=\begin{cases}  \epsilon, &\text{if}\ f(x)\geq\epsilon\\  0, &\text{if}\ f(x)<\epsilon.  \end{cases}
Then 0\leq s(x)\leq f(x). Thus \int_X f(x)\,d\mu\geq\int_X s(x)\,d\mu=\epsilon\mu(\{x\in X: f(x)\geq\epsilon\}). Dividing both sides by \epsilon>0 gives the result.

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