If a random variable has a moment generation function $M(t) = a + (1-a) \, e^t$ with $ 0 < a < 1$
a) Determine the Distribution of $X$
b) Show that the moment of r-class of $X$ is equal to $E[X^r] = 1-a$ , for $1,2,\ldots$
Edit:
So this is my best best approach till now; We know that $M_x(t) = \mathbb{E} \left[e^{tx} \right] = \int_{ -\infty}^{+\infty} e^{t x} f(x) dx $
Also I know that $M_x(t=0)=1$.
So, if I could find $f(x)$, I could easily then find $F(x)$ (distribution).
The problem is that I am doing cyrcles all over my papers .
Should I find $\text{Var}(X)$? thats easy... and then try from this?
Can't think of anything else.