Moment generating function, determine distribution If $Y$ is a random variable such that $\mathrm{E}[Y^k] = \frac{1}{4} + 2^{k-1}$ for $k = 1,2,{...},$ I'm supposed to determine the distribution. I know the answer is $P(X=0) = \frac{1}{4}, P(X=1) = \frac{1}{4}, P(X=2) = \frac{1}{2},$ but I don't know how to reach that answer.
So $\mathrm{E}[Y] = \frac{5}{4}$ and $\mathrm{E}[Y^2] = \frac{9}{4},$ but that is about as far as I've made it.
Is it all about guessing and confirming by computing the moments or is there a method to this?

Knowing those moments, the moment generating function is then $$M_X(t) = 1 + (\tfrac{1}{4} + \tfrac{1}{2} 2^{1})t+(\tfrac{1}{4} +\tfrac{1}{2} 2^{2})\frac{t^2}{2!}+(\tfrac{1}{4} + \tfrac{1}{2}2^{3})\frac{t^3}{3!}+\cdots $$ 
and rewriting $1 = \tfrac{1}{4} +(\tfrac{1}{4} +\tfrac{1}{2} 2^{0})t^0$ we have $$M_X(t)= \tfrac{1}{4}+\tfrac{1}{4}e^t+\tfrac{1}{2}e^{2t}.$$ 
Since $M_X(t) = E\left[e^{tX}\right]$, this is clearly the moment generating function of a discrete random variable $X$ with $P(X=0)=\frac14$, $P(X=1)=\frac14$, $P(X=2)=\frac12$ 
