Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2020 / Paper 1 / 16 / b

Let $N$ count rotten pieces. Each item supplies an independent Bernoulli trial with success probability $p$, so $N$ has a binomial distribution with parameters $10,p$. There are $\left(\genfrac{}{}{0ex}{}{10}{n}\right)$ choices of the rotten items, and each such choice has probability $p^n(1-p{)}^{10-n}$. HenceFor $p=0$ or $p=1$, interpret this as the corresponding distribution concentrated at zero or ten; no ambiguous $0^0$ evaluation is needed.

Write $N={\sum }_{j=1}^{10}{X}_j$, where the independent Bernoulli random variables indicate rottenness. Since $X_j^2=X_j$, their variances are $E[X_j^2]-E[X_j{]}^2=p-p^2$. The independence makes all cross covariances zero, soThis standard deviation vanishes at $p=0$ and $p=1$, when the count is certain.