Past exam of the year 1 of the computer science course of the University of Oxford / 2015 / Discrete mathematics, continuous mathematics, and probability / 8 / b / iv / Solution

Both sides of (ii) are polynomials in the colour count, and they agree at every integer $x\ge 2$. A nonzero polynomial has only finitely many roots, so this is a polynomial identity and holds also at $x=0,1$. It can therefore be evaluated at every possible value of the Poisson random variable $X$.

Take expectations and use the factorial moments from (a)(iv). The sum over $k$ is finite, so linearity of expectation givesCombining this with (iii) proves Dobiński's formula for the Bell numbers:For completeness, the empty set has one partition of a set, giving $B(0)=1$; that case uses the empty-product conventions $0^0=1$ in this identity and a $k=0$ term in the partition sum.