Weighted Importance in sampled data statistics

[X89codered89X]

I was just wondering how exactly to appropriately modify the 1st and 2nd order stats when you want to weight a given sample more heavily. If $\vec{X}$ is my vector of N samples and I have a weight vector $\vec{W}$ of the same dimension, which ideally is measuring the reliability of each sample from 0 to 1. Could I calculate mean and variance using...

$\vec{Y}_i= \vec{X}_i\vec{W}_i <br /> \\<br /> <br /> \mu_{x-weighted} = E[\vec{Y}]<br /> \\<br /> <br /> \sigma_{x-weighted} = E[(\vec{Y}-E[\vec{Y}])^2]<br /> \\<br />$

Let me know what you think. Thanks.

EDIT: $\LaTeX$ mods...

 

[Number Nine]

The weighted mean is given by...

$$\mu_w = \frac{\sum x_iw_i}{\sum w_i}$$

The weighted sample variance is...

$$\sigma^2_w = \frac{\sum w_i(x_i - \mu_w)^2}{\sum w_i}$$

 

[X89codered89X]

Hey thanks!