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Probability - Bernoulli stuff

  1. Nov 3, 2006 #1

    quasar987

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    Consider a sequence of Bernoulli trials where we play until the rth success is attained. Denote [itex]\Omega[/itex] the fundamental set.

    We define a function P on [itex]\Omega[/itex] by saying say that an elementary event that is a k-tuple has a probability of occurence of

    [tex]q^{k-r}p^r[/tex]

    because the trials are independant and the probability of success at each of them is p and the probability of failure is q.

    Now I ask wheter or not with these probabilities assigned to each elementary event, we do have [itex]P(\Omega)=1[/itex]? Did I miss something and it is implied that [itex]P(\Omega)=1[/itex], or we have to show that

    [tex]\sum_{k=r}^{\infty}\binom{k-1}{r-1}q^{k-r}p^r=1[/tex]

    to show that P defined above is indeed a probability on [itex]\Omega[/itex]?
     
    Last edited: Nov 3, 2006
  2. jcsd
  3. Nov 6, 2006 #2
    that looks like a negative binomial distribution... not sure. but you should probably prove it sums to 1.
     
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