Past exam of the mathematics course of the University of Cambridge / 2014 / iii / Paper 35 / 2 / e

Start with the Beta distribution draw in the Beta-binomial exchangeable coupling. Write $s=\alpha +\beta$ andThese are the expected value and variance of $p_M$. The fixed $n$ in this coupling is a dependence parameter chosen by the investigator, not necessarily the observed number of cups. The four numbered stages together address the single printed correlation request.

Conditional on $p_M$, the auxiliary binomial distribution gives $\mathbb{E}[X/n\mid p_M]=p_M$ and $\mathrm{Var}(X/n\mid p_M)=p_M(1-p_M)/n$. Thus large $n$ makes $X/n$ close to $p_M$. The law of total variance also givesThe latent variable encodes the first draw increasingly accurately.

The next Beta distribution has conditional expected value and varianceFor large $n$, its expected value approaches $X/n$, while its variance is at most $1/[4(s+n+1)]$. The final draw therefore stays close to the first. These are prior distributions for the cup probabilities: the auxiliary count is a device for constructing dependence, rather than additional observed tea data.

The law of iterated expectation yieldsOne may obtain $\mathrm{Var}(p_T)=v$ either from its unchanged marginal distribution proved below or directly from the law of total variance: the variance of its conditional mean is $nv/(s+n)$ and its expected conditional variance is $sv/(s+n)$. Hence the correlation coefficient is exactlyAlso $\mathbb{E}[(p_T-p_M{)}^2]=2vs/(s+n)\to 0$. The two probabilities become close while keeping their original beta marginals.