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

  • #1

Main Question or Discussion Point

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.
 

Answers and Replies

  • #2
statdad
Homework Helper
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Notice that

[tex]
X_1 - \bar X = (1-\frac 1 n) X_1 - \frac 1 n \sum_{i\ge2}X_i
[/tex]

and all of [itex] X_1 [/itex] and [itex] X_2, \dots, X_n [/itex] are independent.

* Get the distribution of

[tex] (1 - \frac 1 n) X_1
[/tex]

as well as that of

[tex]
\frac 1 n \sum_{n\ge2} X_i
[/tex]

These are independent as well, so find the distribution of their difference.
 
  • #3
I have Xi-Xbar ~ N(0, (1+1/n)sigma2) ?
 
  • #4
statdad
Homework Helper
1,495
35
Are you sure the variance is

[tex]
\left(1 + \frac 1 n \right) \sigma^2
[/tex]
 

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