# Mathematical Statistics Lesson of the Day – Chebyshev’s Inequality

September 3, 2014 Leave a comment

The **variance** of a random variable is just an **expected value** of a function of . Specifically,

.

Let’s substitute into Markov’s inequality and see what happens. For convenience and without loss of generality, I will replace the constant with another constant, .

Now, let’s substitute with , where is the** standard deviation** of . (I can make this substitution, because is just another constant.)

This last inequality is known as **Chebyshev’s inequality**, and it is just a special version of Markov’s inequality. In a later Statistics Lesson of the Day, I will discuss the motivation and intuition behind it. (*Hint: Read my earlier lesson on the motivation and intuition behind Markov’s inequality*.)

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