# Why is this not a valid moment generating function?

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

Attempt: I tried writing $\frac{t}{1-t}$ as the sum of an infinite series so $\sum t^n$ from $n=1$ to $\infty$. We know the formula for a moment generating function is $\sum e^{tx}f(x)$. So I compared the two and tried to show this leads to a density that does not integrate to 1, but I get that: $$f(n) = \sum (\frac{t}{e^t})^n$$, which in general is convergent since $t<e^t$. How else would I show this?

Thanks!

## 1 Answer

There's no playing about with series needed, no worrying about integrals (well not directly). This is "think about what MGFs are" kind of question.

Consider the definition of an MGF -- $\quad M_X(t)=E(e^{tX})$.

Given that definition, what should $M_X(0)$ be?

Do you notice the problem?

• Got it! Yes of course - this just struck me as a minute after I posted the question. Thanks :) – akeenlogician Sep 26 '16 at 3:30