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Math Help - Binomial Distribution Proving.

  1. #1
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    Binomial Distribution Proving.

    (i) Suppose X has a binomial distribution with probability of success p over n trials. Show that

    E(X) \left[\frac{1}{1 + X}\right] = \frac{1 - (1-p)^{n+1}}{p(n+1)}

    (ii) Suppose now X has a negative binomial distribution with density

     P(X=k) = \left(\begin{array}{cc}k-1\\n-1\end{array}\right) p^nq^{k-n}

    with q = 1-p and k \geq n. Prove that for any function f(x),

    E[qf(X)] = E \left[\frac{(X-n)f(X-1)}{X-1}\right]
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  2. #2
    MHF Contributor matheagle's Avatar
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    For the first one I would rewrite  {X\over X+1}={X+1-1\over X+1}=1-{1\over X+1}

    and obtain 1-E\biggl({1\over X+1}\biggr) instead.

    That can be done by substituting y=x+1 in your sum.
    Last edited by matheagle; April 19th 2009 at 07:48 AM.
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  3. #3
    Moo
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  4. #4
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    Quote Originally Posted by Moo View Post
    Thread closed.
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