What is correct definition of moment generating function.

Question

I had searched the literature and found two different definition of Moment Generating Function (MGF).

$$\phi_{w}(-s)=\int_0^{\infty}\text{exp}(-sw)f(w)\text{d}w \tag{1} $$

$$\phi_{w}(s)=\int_0^{\infty}\text{exp}(sw)f(w)\text{d}w \tag{2} $$

Out of these, which one is correct definition. Any help in this regard is highly appreciated.

Answer 1

both are wrong. The correct definition is the following

$$M_X(t)=\mathbb{E}[e^{tX}]$$

when the expectation exists... not necessarily the expectation is an integral

for example, let $X\sim B(1;\theta)$, that is a bernoulli with parameter $\theta$, its MGF is given by

$$M_X(t)=e^0\cdot (1-\theta)+ e^t\cdot \theta=(1-\theta)+e^t\cdot \theta$$