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Expected Maximum

  1. Sep 25, 2008 #1
    I swear that I used to know this.

    If you have an independent sample of size n, from the uniform distribution (interval [0,[tex]\theta[/tex]]), how do you find the Expected Value of the largest observation(X(n))?
     
  2. jcsd
  3. Sep 25, 2008 #2

    statdad

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    Homework Helper

    If [tex] X_{(n)}[/tex] is the maximum in the sample, you first find its distribution. Since you have a random sample of size [tex] n [/tex], you can write

    [tex]
    F(t) = \Pr(X_{(n)} \le t) = \prod_{i=1}^n \Pr(X_i \le t) = \left(\frac{t}{\theta}\right)^n
    [/tex]

    Differentiate this w.r.t. [tex] t [/tex] to find the density [tex] f(t) [/tex], and the expected value is

    [tex]
    \int_0^{\theta} t f(t) \, dt
    [/tex]
     
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