Answer:
See the proof below.
Step-by-step explanation:
For this case we just need to apply properties of expected value. We know that the estimator is given by:
And we want to proof that
So we can begin with this:
And we can distribute the expected value into the temrs like this:
And we know that the expected value for the estimator of the variance s is , or in other way
so if we apply this property here we have:
And we know that so using this we can take common factor like this:
And then we see that the pooled variance is an unbiased estimator for the population variance when we have two population with the same variance.