Chebyshev's Inequality:

We discussed in class the "famous" inequality, discovered
by the mathematician Chebyshev, which relates the mean and
the variance:

Suppose X is an RV with mean u, i.e.,

                      E(X) = u 

and variance s^2, (read this as "s squared") i.e.,

                     V(X) = s^2

so that the standard deviation is

                     D(X) = s

Then the following is true:

Pick any number b, then the probability that X takes
a value a distance greater than or equal to b from the
mean, is AT MOST (s^2)/(b^2).

Or as a formula

                      P(|X - u| >= b) <= (s^2/b^2)

Problem: Suppose the IQ scores of 1 million people have
         a mean of 100 and a standard deviation of 10.

         Find an upper bound on the number of people with
         a score greater than or equal to 130.

Solution: So, u = 100, s = 10, so s^2 = 100.
          
          We are interested in the number of people with a
          score of at least 130, so that the DISTANCE of their
          score from the mean must at least 30. I.e., we want
   
                       P([X-100] >= 30)

          Now notice that

             P([X-100] >= 30) <=  P(|X-100| >= 30)

          This is because P(|X-100| >= 30) includes scores that are
          less than or equal to 70 as well!
 
          Now we can apply Chebyshev's inequality to get

      P([X-100] >= 30) <=  P(|X-100| >= 30) <= (100/900) = 1/9