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I am learning about moment-generating functions and need a little help with this exercise:

Let's say we have a moment-generating function for a random variable X, $M_X(t)=(\frac{0.25e^t}{1-0.75e^t})^4\cdot e^{-2t}$ for $t<-ln0.75$. How would we find $X$ from this?

Thanks!

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    My first thought was that the variable could be broken into a sum of 4 negative binomial distributions plus.. something, but I'm not sure what that something is.2012-02-01

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Hint: Your textbook probably has a list of common moment generating functions that you can check. Since you mention it, if $Y$ is negative binomial with parameters $r$ and $p$ it has moment generating function $m_Y(t)=\left[{pe^t\over 1-(1-p)e^t}\right]^r.$ Do you see how to match that up with part of your expression? How can you account for the extra $e^{-2t}$?

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    I'll let you take it $f$rom here.2012-02-01