Chapter 9.5, Problem 65E

In this exercise, we illustrate the direct use of the Rao–Blackwell theorem. Let Y₁, Y₂, …, Y_n be independent Bernoulli random variables with

$p(y_i|p)=p^{yi}{(1-p)}^{1-yi},y_i=0,1$

That is, P(Y_i = 1) = p and P(Y_i = 0) = 1 − p. Find the MVUE of p(1 − p), which is a term in the variance of Y_i or $W={\displaystyle {\sum }_{i=1}^n{Y}_i}$, by the following steps.

a Let $T=\{\begin{array}{l}1,\text{if}{Y}_{\text{1}}=\text{1 and}{Y}_{\text{2}}=0,\\ 0,\text{otherwise}.\end{array}$

Show that E(T) = p(1 − p).

b Show that

$\begin{array}{l}P(T=1|W=w)=\frac{w(n-w)}{n(n-1)}\\ \end{array}$

c Show that

$E(T|W)=\frac{n}{n-1}\left[\frac{W}{n}\left(1-\frac{W}{n}\right)\right]=\frac{n}{n-1}\bar{Y}(1-\bar{Y})$

and hence that $n\bar{Y}(1-\bar{Y})/(n-1)$ is the MVUE of p(1 − p).