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.)