When is the following unbiased?

[DotFourier]

Homework Statement

Let $$S_{1}^2, S_{2}^2, S_{3}^2$$ be sample variances of sample size $$n_{1}, n_{2}. n_{3}$$ respectively. The populations have means $$\mu_{1}, \mu_{2}, \mu_{3}$$ respectively with common variance $$\sigma^2.$$

When is $$\pi_{1}S_{1}^2 + \pi_{2}S_{2}^2 + \pi_{3}S_{3}^2$$
unbiased?

Homework Equations

The Attempt at a Solution

I was under the impression that the sample variance was always unbiased. Wouldn't the expected value of the above expression always yield the same expression?[/B]

 

[mfb]

How are the S defined? They might be biased estimators for σ², then you have to account for that.