I want to compute the moment generating function of a random variable X that has a PDF defined only for $x\ge 0$. I have found some papers that use the following definitions for the MGF: \begin{equation} \text{MGF}_X(s)=\mathbb{E}(e^{-sX})=\int_0^\inf e^{-sx}f(x)dx \end{equation}
How is this minus sign came ?? By definition, the MGF is expressed as $\mathbb{E}(e^{sX})$, meaning without a minus. Are they some conditions on X such that MGF$_X =\mathbb{E}(e^{-sX})$ becomes correct and valid ?
If so, I want to apply the lotus principle on the minus form as it makes it easier to bound and integrate. Any suggestions ??