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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?


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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. – ro44 Feb 1 '12 at 1:22
up vote 1 down vote accepted

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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Yeah, that's the thing. Like I mentioned in the comments I thought about using negative binomial distributions. But I don't know what to do with the cursed $e^{-2t}$ there. – ro44 Feb 1 '12 at 1:39
I will proceed slowly with hints: $e^{-2t}$ itself is a moment generating function. Of what kind of random variable? – Byron Schmuland Feb 1 '12 at 1:42
I can match it with a normal distribution, but can you combine discrete and continuous random variables like that? – ro44 Feb 1 '12 at 1:44
No it is easier. The normal mgf has $t^2$ in the exponent, not $t$. Try a constant random variable. – Byron Schmuland Feb 1 '12 at 1:56
Oh wait, perhaps $Z=-2$? I thought about using that earlier but got confused. – ro44 Feb 1 '12 at 2:00

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