Re: [機統] 條件機率相關問題

[Enas (快要畢業了)]

there is another manner that we can apply :


M (t) = E(exp (tX)|U=p) = [ q + p exp(t) ] ^n  , q = 1-p
 X|U

M (t) = EE(exp(tX|U))
 X
      = E( (1-U) + Uexp(t))^n )

      = E(Σ(n)(1-U)^n (Uexp(t))^n-k )
            (k)
Also ,  E((1-U)^n (Uexp(t))^n-k) = Beta ( k+1 , n+1-k )

use this result , we have


M (t)
 X
   n! k! (n-k)!  exp(nt) exp(-kt)
=Σ------------------
   k! (n-k)! (n+1)!

          exp(nt)
= -------------------------
     n+1 (1 - exp(-t)

thus we know X~Discrete Uniform(0,1,2 ... n)

--------------------------------------------

have fun  :)

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推 ethan0221   :感謝! :]