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Math Help - Poisson distribution - help with question!

  1. #1
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    Poisson distribution - help with question!

    Hello,

    I am having problems solving the following question:
    Let the random variable X have a Poisson distribution with parameter [lambda]. Show that for every n>=1 one has E(X^n) = [lambda]E[(X +1)^(n-1)]: Note that, using this formula, we can nd E(X);E(X2), . . .recursively.

    I have no idea what to do... Please help

    -Thanks
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  2. #2
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    Quote Originally Posted by AAZZ View Post
    Hello,

    I am having problems solving the following question:
    Let the random variable X have a Poisson distribution with parameter [lambda]. Show that for every n>=1 one has E(X^n) = [lambda]E[(X +1)^(n-1)]: Note that, using this formula, we can nd E(X);E(X2), . . .recursively.

    I have no idea what to do... Please help

    -Thanks
    \displaystyle<br />
\sum_{x = 0} ^ \infty x^n \frac{\lambda^x e^{-\lambda}}{x!} = \sum_{x = 0} ^ \infty (x + 1)^n \frac{\lambda^{x + 1} e^{-\lambda}}{(x + 1)!}.<br />

    Fill in the gaps and complete the problem.
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  3. #3
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    What do we do with this formula??
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  4. #4
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    Quote Originally Posted by stats2010 View Post
    What do we do with this formula??
    The expectation (ha ha, ... expectation) is that you realise that \displaystyle \sum_{x = 0} ^ \infty (x + 1)^n \frac{\lambda^{x + 1} e^{-\lambda}}{(x + 1)!} = \lambda \sum_{x = 0} ^ \infty (x + 1)^{n-1} \frac{\lambda^{x} e^{-\lambda}}{x!} ....
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  5. #5
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    Quote Originally Posted by AAZZ View Post
    Hello,

    I am having problems solving the following question:
    Let the random variable X have a Poisson distribution with parameter [lambda]. Show that for every n>=1 one has E(X^n) = [lambda]E[(X +1)^(n-1)]: Note that, using this formula, we can nd E(X);E(X2), . . .recursively.

    I have no idea what to do... Please help

    -Thanks
    This appears to be part of an assignment that might count towards your final grade. Please see rule #6: http://www.mathhelpforum.com/math-he...ng-151424.html
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