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Probability generating function.

  1. Nov 3, 2012 #1
    For any integer valued RV X Summation n=0 to infinity of s^n P(X=<n) = (1-s)^-1 * Summation k=0 to infinity of P(x=k)s^k

    Why does Sum k=0 to infinity P(x=k)s^k = sum n=0 to infinity of P(X=< n)
  2. jcsd
  3. Nov 3, 2012 #2


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    Staff: Mentor

    $$\sum_{n=0}^{\infty} s^n P(X\leq n) = \sum_0^{\infty} s^n \sum_{k=0}^n P(X=k)$$
    You can re-write this double sum and use k as primary index. The inner sum (over n) has an explicit formula then, and you get the correct result.
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