Past exam of the year 1 of the computer science course of the University of Oxford / 2016 / Discrete mathematics, continuous mathematics and probability / 8 / c / ii / Solution

A Bernoulli random variable with parameter $1/2$ has $G_Y(s)=(1+s)/2$. Part (i) and the geometric distribution formula giveComparing coefficients in the probability generating function identifiesThus $X$ has the zero-based geometric distribution with success parameter $2/3$. The probabilities sum to one, and its mean $1/2$ agrees with the random-sum mean $\mathbb{E}(N)\mathbb{E}(Y_1)=1\cdot 1/2$.