Past exam of the year 1 of the computer science course of the University of Oxford / 2018 / Discrete mathematics, continuous mathematics, and probability / 9 / a / ii

Independence gives the probability generating function of the sum as the product:This is the probability generating function of the Poisson distribution with parameter $\mu +\lambda$. Uniqueness of the coefficients of a power series therefore gives $X+Y\sim \mathrm{Poisson}(\mu +\lambda )$. The product rule for these generating functions requires the stated independence.