31. An estimator is said to be consistent if for any ! 0, P( ...

31. An estimator ˆ

 is said to be consistent if for any ! 0, P(⏐ˆ

  ⏐  !) 0 0 as n 0 . That is, ˆ

 is consistent if, as the sample size gets larger, it is less and less likely that ˆ

 will be further than ! from the true value of . Show that X is a consistent estimator of when 2   by using Chebyshev’s inequality from Exercise 44 of Chapter 3. [Hint:

The inequality can be rewritten in the form P(⏐Y  Y⏐ !)  2 Y /!

Now identify Y with X.]