Question

The variance plays an integral role in understanding a distribution,

a) Why does a function having a finite variance, i.e. a variance exist, imply that the mean is also finite?

b) Previously, we explained how the sample variance is both an unbiased and consistent estimator of the population variance, and how logically since s_n² converges in probability to var², then so does the standard deviation, i.e. s_n converges in probability to var. Explain why.

Hint: You should be using a theorem to explain all of this.

Answer 1

a) Since for any Variance is given by

Where E stands for expectation and is the mean

So we can write mean as

Which implies that if is finite, then is finite that is mean is finite

b) Its given that   converges in probability to

So,

Now, we know