Past exam of the year 1 of the computer science course of the University of Oxford / 2013 / Discrete mathematics, probability and continuous mathematics / 7 / a / ii / Solution

For the binomial distribution, the binomial theorem givesDifferentiation of the probability generating function recovers the first expected value and second factorial moment:Since $X^2=X(X-1)+X$, the variance isThusThe formulas also hold for $n=0,1$ and the degenerate choices $p=0,1$, with derivatives read directly from the resulting polynomial.