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

  1. Mar 31, 2010 #1
    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.
     
  2. jcsd
  3. Mar 31, 2010 #2

    statdad

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    Homework Helper

    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.
     
  4. Mar 31, 2010 #3
    I have Xi-Xbar ~ N(0, (1+1/n)sigma2) ?
     
  5. Mar 31, 2010 #4

    statdad

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    Homework Helper

    Are you sure the variance is

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