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Math Help - Poisson Distribution

  1. #1
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    Poisson Distribution

    Let X be a Poisson distribution. Show that P\{X=k\} increases as it approaches to  \lambda , reaching it's max when k is the largest integer less then \lambda then decrease as k > \lambda

    Hint: Consider \frac{P\{X=k\}}{P\{X=k-1\}}.

    since the Poisson distribution is defined as:

    \frac{\lambda^k}{k!}e^{-k} I was thinking of using derivatives to max out the equation with respect to k, but realized that due to the factorial I can't since it's not continuous.

    I'm not sure I'm doing this right, but if I apply the hint I would get:

     \left( \frac{\lambda^k}{k!}e^{-k} \right) \left( \frac{k!}{\lambda^k}e^{-k} \right) = \left( \frac{(k-1)!}{k!} \right) \left( \frac{\lambda^k}{\lambda^{k-1}} \right) = \frac{\lambda}{k}

    now I'm not sure what I have to do
    Last edited by lllll; September 14th 2008 at 10:48 PM.
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by lllll View Post
    Let X be a Poisson distribution. Show that P\{X=k\} increases as it approaches to  \lambda , reaching it's max when k is the largest integer less then \lambda then decrease as k > \lambda

    Hint: Consider \frac{P\{X=k\}}{P\{X=k-1\}}.

    since the Poisson distribution is defined as:

    \frac{\lambda^k}{k!}e^{-k} I was thinking of using derivatives to max out the equation with respect to k, but realized that due to the factorial I can't since it's not continuous.

    I'm not sure I'm doing this right, but if I apply the hint I would get:

     \left( \frac{\lambda^k}{k!}e^{-k} \right) \left( \frac{k!}{\lambda^k}e^{-k} \right) = \left( \frac{(k-1)!}{k!} \right) \left( \frac{\lambda^k}{\lambda^{k-1}} \right) = \frac{\lambda}{k}

    now I'm not sure what I have to do
    \frac{P\{X=k\}}{P\{X=k-1\}}=\left( \frac{(k-1)!}{k!} \right) \left( \frac{\lambda^k}{\lambda^{k-1}} \right) = \frac{\lambda}{k}.

    Therefore:

    \frac{P\{X=k\}}{P\{X=k-1\}}\ge 1 when k\le \lambda

    and:

    \frac{P\{X=k\}}{P\{X=k-1\}} < 1 when k > \lambda

    RonL
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