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Negative binomial probability function question

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  • Negative binomial probability function question

    There are two ways to express the negative binomial probability function where the "combination" can be expressed differently.

    (r + x - 1 choose x) is equivalent to (r + x - 1 choose r -1)
    I know I can just accept that there are two ways to express the function but I am trying understand how they are the same.
    Does this mean that x = r -1?

  • #2
    First, the answer is no.

    Simply expand these two expressions, you get:

    (r + x - 1 choose x) = (r + x - 1)!/ [x! *(r -1)!]
    (r + x - 1 choose r-1) = (r + x - 1)!/ [(r -1)! * x!]

    One is number of successes before the final success you want.
    Another one is number of failures before the final success.

    In either way, you have r + x trials and the last one is the success.

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    • #3
      Oh ok. Thanks for the reply dude! This makes sense.

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