# Thread: Biased/Unbiased help

1. ## Biased/Unbiased help

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

2. Let's say we want to know if a statistic Y is a biased/unbiased estimator for $\displaystyle \theta$.

If $\displaystyle E(Y_1)=\theta$ then Y is an unbiased estimator of $\displaystyle \theta$

Now if $\displaystyle E(Y_2) = n \theta$ then Y is a biased estimator of $\displaystyle \theta$.

$\displaystyle E(\frac{Y_2}{n}) = \theta$ then Y2 is an unbiased estimator of $\displaystyle \theta$

I'm assuming you need to determine if a statistic is an unbiased estimator of $\displaystyle \mu$ since it's the only unknown variable you have.