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Math Help - hard proof- gamma distribution

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    hard proof- gamma distribution

    if X is an exponential random variable with mean 1/lambda, show that the expected value of x^k = k!/lambda^k
    Hint: make use of the gamma density function
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    The density function for an exponential is

    f(x)=\lambda e^{-\lambda^x}

    so

    \mathbb{E}[x^k]=\int_{0}^{\infty}x^{k}f(x)dx =\lambda\int_{0}^{\infty}x^{k}e^{-\lambda x}dx

    let u=\lambda x and the resulting integral is the gamma function.

    \Gamma(k+1=k!)

    Gamma function - Wikipedia, the free encyclopedia
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