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.