Does an MGF $M(t)$ exist if it is finite only at $t=0$?
I read that an MGF exists if it is finite on some open interval $(-a,a)$ containing $0$. I'm not sure if $0$ alone counts as such an open interval!
The reason I ask this question is because I'm working through an example which asks if the moment generating function of a Weibull distribution with $\lambda=1$ and $\gamma=\frac{1}{3}$ exists.
Let $T=X^3$ with $X \sim Exp(1)$. Then
$$ \mathbb{E}\left[ e^{tT} \right] = \mathbb{E}\left[ e^{tX^3} \right] = \int_0^{\infty} e^{tx^3-x}\ dx. $$
The example says that this integral diverges for $t>0$, so the MGF of $T$ does not exist. However, I think that the integral does not diverge for $t\leq 0$, and I would expect that this would be enough for the MGF to exist.
Could anybody shed some light on this please?