Chapter 6.1, Problem 17E

In Chapter 3, we defined a negative binomial rv as the number of failures that occur before the rth success in a sequence of independent and identical success/failure trials. The probability mass function (pmf) of X is

$nb(x;r\text{, }p)=\left(\begin{array}{l}x\text{ + }r\text{ - 1}\\ x\end{array}\right)p^r{(1\text{ - }p)}^{x}x\text{ = 0, 1, 2, }\mathrm{...}$

1. a. Suppose that r > 2. Show that

   $\hat{p}\text{ = (}r\text{ - 1)/(}X\text{ + }r\text{ - 1)}$

   is an unbiased estimator for p. [Hint: Write out E( $\hat{p}$) and cancel x + r - 1 inside the sum.]

2. b. A reporter wishing to interview five individuals who support a certain candidate begins asking people whether (S) or not (F) they support the candidate. If the sequence of responses is SFFSFFFSSS, estimate p = the true proportion who support the candidate.