I have read in a probability book that the advantage of dealing with Laplace transform (for non-negative random variable) rather than moment generating function is that Laplace transform is always between $0$ and $1$. I don't understand what is the advantage ?
Laplace transform is $\phi(-t)$ whereas moment generating function is $\phi(t)$
Best Answer
Moment generating functions are also called two sided Laplace transform. In that, you can say for a random variable $X$, the Laplace transform can be $E[e^{-sX}]$ for $s \geq 0$. The reason that Laplace transform is between $0$ and $1$ is that $0\leq e^{-sX} \leq 1$.
There are obvious advantages of Laplace transforms if you have ever taken a Linear Systems class. Read this to gain more insight.