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Math Help - Determine distribution, Prove Laplace Transform

  1. #1
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    Determine distribution, Prove Laplace Transform

    Consider a game in which a fair coin is tossed indefinitely. Every time heads appears you move 1 metre to the right, and if tails you stay where you are. You start at position 0. Let  T_n be the number of tosses needed to first enter position n.

    By definition  T_0 = 0

    Determine the distribution of  T_1 and prove that the Laplace Transform of  T_n is  \mathbb{E} e^{-sT_n} = (2e^s - 1)^{-n}, s \geq 0
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    Could someone tell me if the distribution of  T_1 is a geometric distribution?
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  3. #3
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    Quote Originally Posted by woody198403 View Post
    Could someone tell me if the distribution of  T_1 is a geometric distribution?
    Yes, it is.
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    So the distribution of  T_1 is geometric.
    But the distribution of  T_n is not geometric, right?
    I am having trouble determining the distribution of  T_n . My original thought was that it was perhaps the exponential distribution because the exponential distribution can be considered the continuous version of the geometric distribution, and unless I have made an error in my calculations, the Laplace Transform of the exponential dist. is not the same as the solution given in the question.

    Just to double check though, the Laplace transform of a continuous random variable X is
     L(s) = \mathbb{E} e^{-sX} = \int_0^{\infty} e^{-sx} f(x) \ dx where  f(x) is the pdf of  X ??
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    Quote Originally Posted by woody198403 View Post
    So the distribution of  T_1 is geometric.
    But the distribution of  T_n is not geometric, right? Mr F says: Correct. The distribution of  {\color{red}T_n } is not geometric. It has a negative binomial distribution.

    I am having trouble determining the distribution of  T_n . My original thought was that it was perhaps the exponential distribution because the exponential distribution can be considered the continuous version of the geometric distribution, and unless I have made an error in my calculations, the Laplace Transform of the exponential dist. is not the same as the solution given in the question. Mr F says:  {\color{red}T_n } is clearly a discrete random variable and therefore cannot have a continuous distribution.

    Just to double check though, the Laplace transform of a continuous random variable X is
     L(s) = \mathbb{E} e^{-sX} = \int_0^{\infty} e^{-sx} f(x) \ dx where  f(x) is the pdf of  X ??
    ..
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  6. #6
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    Thank-you Mr F. I dont know why I was looking at continuous distributions . Your help is greatly appreciated.
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  7. #7
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    Re: Determine distribution, Prove Laplace Transform

    This question really intrigues me, I understand to find the Laplace Transform of Tn, you integrate Tn x exp(-st) from the limits of infinity to zero { ∫∞^0 T n e^(-st) }Please excuse the bad formatting. But based on this question I am confused as to how to find the function of Tn. Could someone please explain?

    Thanks
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