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Math Help - Probability generating functions - please spot my mistake!

  1. #1
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    Probability generating functions - please spot my mistake!

    According to Zeno (495-435 BC) there was a race in which Achilles had to catch a tortoise having given it a start. Suppose that the race began with Achilles waiting on the starting line while the tortoise was given a start of r time units, where r is a positive integer. Suppose also
    that, during each time unit, the tortoise either moves 1 metre with probability p and stays where
    it is with probability q (p + q) = 1. Write down the distribution of the distance travelled by the tortoise
    before it rst fails to move and fi nd the probability generating function.

    My answer: geometric, parameter p

    Generating function: qs/(1-ps)
    Once he has started, Achilles runs at a constant rate of 1 metre per time unit.

    Find the probability generating function for the distance from the start at which Achilles catches the tortoise and show that its expectation is rp/q

    My answer: Find where tortoise has not moved r times

    Binomial distribution, parameter q

    generating function: (p + qs)^n

    I have put my first genearting function is this one, and differentuiated, and put s = 1 to give just n, so I mustve made a mistake

    Many Many Thanks - I've been stuck on this for days!
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  2. #2
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    Quote Originally Posted by James0502 View Post
    According to Zeno (495-435 BC) there was a race in which Achilles had to catch a tortoise having given it a start. Suppose that the race began with Achilles waiting on the starting line while the tortoise was given a start of r time units, where r is a positive integer. Suppose also
    that, during each time unit, the tortoise either moves 1 metre with probability p and stays where
    it is with probability q (p + q) = 1. Write down the distribution of the distance travelled by the tortoise
    before it rst fails to move and fi nd the probability generating function.

    My answer: geometric, parameter p

    Generating function: qs/(1-ps)
    Once he has started, Achilles runs at a constant rate of 1 metre per time unit.

    Find the probability generating function for the distance from the start at which Achilles catches the tortoise and show that its expectation is rp/q

    My answer: Find where tortoise has not moved r times

    Binomial distribution, parameter q

    generating function: (p + qs)^n

    I have put my first genearting function is this one, and differentuiated, and put s = 1 to give just n, so I mustve made a mistake

    Many Many Thanks - I've been stuck on this for days!
    I think you may need the negative binomial distribution on the second part, not the binomial distribution.
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