Past exam of the options II course of the University of Oxford / 2011 / M / M4 / b / i / Solution

The binomial series gives, for $|z|<14/5$,All coefficients are nonnegative, and $g(1)=2/3+1/3=1$. Thus defineThese numbers sum to $1$, so they define a probability distribution on the nonnegative integers with the desired probability generating function.