I'm struggling hugely with breaking down my M.G.F. into something that I can use to give me a probability function of $X$, the problem reads:
Find the probability function, $f$, of $X$ including domain given the moment generating function of $X$ is $$M_X(t) = (0.4e^{-t} + 0.6e^{t})^2$$ I know the $0.4$ and $0.6$ values are significant, but I feel powerless as to where to start. Anybody out there with the know how to lend a hand?
Let $X$ be a random variable that takes on value $-1$ with probability $0.4$ and value $1$ with probability $0.6$. Then $X$ has mgf $0.4e^{-t}+0.6e^{t}$.
It follows that our mgf is the mgf of the sum of two independent random variables $X_1$ and $X_2$, each of which has the distribution described in the first paragraph.
Alternately, expand the square. We get $(0.4)^2 e^{-2t}+2(0.4)(0.6)e^{0\cdot t}+(0.6)^2 e^{2t}$. This is the mgf of a random variable that takes on value $-2$ with probability $(0.4)^2$, value $0$ with probability $2(0.4)(0.6)$, and $2$ with probability $(0.6)^2$.