12.1 Binomial and Gaussian distributions

(a) The binomial distribution

In \(n\) independent trials, each succeeding with probability \(p\) (and failing with \(q=1-p\)), the number of successes \(x\) has probability

\[P(X=x)=\binom nx p^xq^{n-x}.\]

Moments. Write \(X=\sum_{i=1}^n\xi_i\), where \(\xi_i=1\) with probability \(p\) and \(0\) with probability \(q\). Each trial has \(\langle\xi\rangle=p\) and \(\langle\xi^2\rangle=p\), so \({\rm Var}(\xi)=p-p^2=pq\). Since the trials are independent, means and variances add: