After searching, I found two questions like mine, but didn't see my answer to my question.
- Finding a probability distribution given the moment generating function
- Finding probability using moment-generating functions
My question is how to find any probability distribution function, given its moment generating function. In particular, how to find this from First Principles (and not memorizing a table).
Let's try an example:
Let $ X \perp Y$. Define the moment generating functions for $X, Y$ respectively as $$M_X(t)=\exp(2e^t-2), M_Y(t)=\left(\frac{3}{4}e^t+ \frac{1}{4}\right)^{10}$$ Find $P(X+Y = 2)$.
First, the problem doesn't tell us whether the distributions are continuous or discrete, so I assume continuous. Now, how do we solve the following for $f_X(x)$?
$$M_X(t)= \int_{-\infty}^{\infty}e^{xt} f_X(x) \ dx = \exp ( 2\ e^t - 2)\tag{1}$$
Next, can we take the derivative with respect to $x$ to both sides, to bring us closer to the solution $f_X(x)$?
I read that a m.g.f. $m_X(t)$ is characteristic to and unique to the distribution of $X$. I saw something about Laplace Transforms in another question, but we have learned nothing of that sort in this course.