Moment-Generating Function question

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SUMMARY

The moment-generating function (MGF) of a random variable, denoted as $M_x(t)$, cannot equal $\frac{t}{t-1}$ because it violates the fundamental property that $M_x(0)$ must equal 1. In the discussion, the user attempted to differentiate the function and found a mean and variance of 1, which is misleading. The critical takeaway is that the MGF must satisfy specific conditions, and in this case, the given function does not meet those requirements.

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Q:Explain why there can be no random variable for which $M_x(t)=\frac{t}{t-1}$
($M_x(t)$ is the moment-generating function of the random variable $x$.)

A: I tried differentiating it twice and got mean of $x=1$ and variance of $x=1$ which seems fine. Maybe its because $M_x(0)$ is not equal to $1$?Thanks in advance for any help with this problem.
 
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JGalway said:
Maybe its because $M_x(0)$ is not equal to $1$?

Yes, that's exactly the reason.
 

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