A sample of normal RVs - the distribution of Xi-Xbar?

[Phillips101]

We have X1,...,Xn~N(mu, sigma2)

The crux of my problem is finding out the distribution of, say, X1-Xbar (where Xbar is the mean of the n RVs). This is going to end up proving the independence of Xbar and Sxx, btw.

I know Xbar~N(mu, sigma2/n), but I don't know how to find the distribution of a difference of normal RVs with different arguments?

Thanks for any help.

 

[statdad]

Notice that

$$X_1 - \bar X = (1-\frac 1 n) X_1 - \frac 1 n \sum_{i\ge2}X_i$$

and all of $X_1$ and $X_2, \dots, X_n$ are independent.

* Get the distribution of

$$(1 - \frac 1 n) X_1$$

as well as that of

$$\frac 1 n \sum_{n\ge2} X_i$$

These are independent as well, so find the distribution of their difference.

 

[Phillips101]

I have Xi-Xbar ~ N(0, (1+1/n)sigma2) ?

 

[statdad]

Are you sure the variance is

$$\left(1 + \frac 1 n \right) \sigma^2$$