There is widespread agreement amongst the managers of the Reliable Motor Company that the number $X$ of faulty cars produced in a month has a binomial distribution

$$P(X=s)=\left(\begin{array}{c} n \\ s \end{array}\right) p^{s}(1-p)^{n-s} \quad(s=0,1, \ldots, n ; \quad 0 \leqslant p \leqslant 1)$$

where $n$ is the total number of cars produced in a month. There is, however, some dispute about the parameter $p$. The general manager has a prior distribution for $p$ which is uniform, while the more pessimistic production manager has a prior distribution with density $2 p$, both on the interval $[0,1]$.

In a particular month, $s$ faulty cars are produced. Show that if the general manager's loss function is $(\hat{p}-p)^{2}$, where $\hat{p}$ is her estimate and $p$ the true value, then her best estimate of $p$ is

$$\hat{p}=\frac{s+1}{n+2}$$

The production manager has responsibilities different from those of the general manager, and a different loss function given by $(1-p)(\hat{p}-p)^{2}$. Find his best estimate of $p$ and show that it is greater than that of the general manager unless $s \geqslant \frac{1}{2} n$.

[You may use the fact that for non-negative integers $\alpha, \beta$,

$$\left.\int_{0}^{1} p^{\alpha}(1-p)^{\beta} d p=\frac{\alpha ! \beta !}{(\alpha+\beta+1) !}\right]$$