I'm working through Schaum's Outline of Probability, Random Variables, and Random Processes, and am stuck on a question about moment-generating functions. If anyone has the 2nd edition, it is question 4.60, part (b).
The question gives the following initial information: $E[X^k] = 0.8$ for k = 1, 2, ...
The moment-generating function for this is the following: $0.2 + 0.8\sum_{k=0}^\infty\frac{t^k}{k!} = 0.2 + 0.8e^t$
The question is asking to find $P(X=0)$ and $P(X=1)$. The answers are given, $P(X=0)=0.2$ and $P(X=1)=0.8$, but I'm not seeing how the book arrived at these answers.
Using the definition of moment-generating functions, I see that the following equation is utilized: $\sum_{i}e^tx_i*p_X(x_i) = 0.2 + 0.8\sum_{k=0}^\infty\frac{t^k}{k!} = 0.2 + 0.8e^t $
But I'm not seeing how the $p_X(x_i)$ is extracted from that equation.
Any help is greatly appreciated. Thanks.