Hi, if someone could take a look through and help me on this problem it would me much appreciated.

Note all the following distributions have sample size 9.

Let $\displaystyle X - N(\mu, 0.3^2)$ and $\displaystyle Y - N(\mu, 0.6^2)$.

We then form the following distributions:

$\displaystyle U = \frac{1}{2}X + \frac{1}{2}Y$

$\displaystyle V = \frac{2}{3}X + \frac{1}{3}Y$

$\displaystyle W = \frac{4}{5}X + \frac{1}{5}Y$

In each of the above cases work out the sampling distribution of the estimator, explain whether it's unbiased and give the standard error.

Here's what I've managed to do so far:

$\displaystyle U - N(\mu, 0.45^2)$

$\displaystyle V - N(\mu, 0.4^2)$

$\displaystyle W - N(\mu, 0.36^2)$

Using the fact that $\displaystyle \bar{X} - N(\mu, \frac{\sigma^2}{n})$, and with a sample size of 9, we can then get the sampling distributions of the 3 estimators:

$\displaystyle \bar{U} - N(\mu, \frac{9}{400})$

$\displaystyle \bar{V} - N(\mu, \frac{4}{225})$

$\displaystyle \bar{W} - N(\mu, \frac{9}{625})$

I think these are right, please correct me if I've made a mistake somewhere.

Now the standard error of the estimator is $\displaystyle \frac{\sigma}{\sqrt{n}}$, giving us

SE $\displaystyle \bar{U} = \frac{3}{20}$

SE $\displaystyle \bar{V} = \frac{2}{15}$

SE $\displaystyle \bar{W} = \frac{3}{25}$

I think this is correct up to here. The only thing I'm really stuck on is how to find whether they are biased or unbiased?

Thanks in advance

Craig