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Math Help - Moment Generating Function of a Discrete Uniform Distr

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    Moment Generating Function of a Discrete Uniform Distr

    I've proven the first part, but I don't seem to know how to take the derivative of this function to find the mean of the distribution.

    The question is If a random variable x has the discrete uniform distribution
    f(x;k) = 1/k, x = 1,2,3,...k and 0 elsewhere, can it be shown that
    the MGF of x is Mx(t) = (e^t(1-e^kt)/k(1-e^t) ? This part I've proven

    We define Mx(t) = E(e^(xt)). Then for the uniform distribution given,
    Mx(t) = ∑▒〖[e^(xt)(1/k)]〗
    = (1/k)[e^t + e^(2t) + e^(3t) + ..... + e^(kt)]
    = (1/k)e^t[1-e^(kt)]/(1-e^t)

    The last part says find the mean of this distribution by evaluating lim t->0 M'X(t).

    So I guess its asking to take the derivative of MGF of x is Mx(t) = (e^t(1-e^kt)/k(1-e^t) but I can't seem to get it to the mean of (K+1)/2

    Sorry if this post is kind of confusing, its because I'm quite confused on how to do this problem. Can someone please lend a hand on that last part? Thanks!
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  2. #2
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    Quote Originally Posted by xuyuan View Post
    I've proven the first part, but I don't seem to know how to take the derivative of this function to find the mean of the distribution.

    The question is If a random variable x has the discrete uniform distribution
    f(x;k) = 1/k, x = 1,2,3,...k and 0 elsewhere, can it be shown that
    the MGF of x is Mx(t) = (e^t(1-e^kt)/k(1-e^t) ? This part I've proven

    We define Mx(t) = E(e^(xt)). Then for the uniform distribution given,
    Mx(t) = ∑▒〖[e^(xt)(1/k)]〗
    = (1/k)[e^t + e^(2t) + e^(3t) + ..... + e^(kt)]
    = (1/k)e^t[1-e^(kt)]/(1-e^t)

    The last part says find the mean of this distribution by evaluating lim t->0 M'X(t).

    So I guess its asking to take the derivative of MGF of x is Mx(t) = (e^t(1-e^kt)/k(1-e^t) but I can't seem to get it to the mean of (K+1)/2

    Sorry if this post is kind of confusing, its because I'm quite confused on how to do this problem. Can someone please lend a hand on that last part? Thanks!
    M = \frac{1}{k} \left( \frac{1 - e^{kt}}{e^{-t} - 1}\right). Differentiate this. Then take the limit t --> 0.

    And if all you want to do is get the mean, it might even be easiest to first substitute the Maclaurin series for e^x and e^kx, simplify the result and then take the derivative and get the limit ....
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