In Chap. 2, we defined a negative binomial rv as the number...

In Chap. $2,$ we defined a negative binomial rv as the number of trials required to achieve the $r$ th success in a sequence of independent and identical success/failure trials. The probability mass function $(\mathrm{pmf})$ of $X$ is $n b(x ; r, p)=\left\{\begin{array}{cc}{\left( \begin{array}{c}{x-1} \\ {r-1}\end{array}\right) p^{r}(1-p)^{x-r}} & {x=r, r+1, r+2, \ldots} \\ {0} & {\text { otherwise }}\end{array}\right.$ $\begin{aligned} \text { (a) Suppose that } r \geq 2 . \text { Show that } & \\ & \hat{P}=(r-1) /(X-1) \end{aligned}$ is an unbiased estimator for $p .[$Hint : Write out $E(\hat{P})$ as a sum, then make the substitutions $y=x-1$ and $s=r-1 . ]$ (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.

In Chap. $2,$ we defined a negative binomial rv as the number of trials required to achieve the $r$ th
success in a sequence of independent and identical success/failure trials. The probability mass
function $(\mathrm{pmf})$ of $X$ is
$n b(x ; r, p)=\left\{\begin{array}{cc}{\left( \begin{array}{c}{x-1} \\ {r-1}\end{array}\right) p^{r}(1-p)^{x-r}} & {x=r, r+1, r+2, \ldots} \\ {0} & {\text { otherwise }}\end{array}\right.$
$\begin{aligned} \text { (a) Suppose that } r \geq 2 . \text { Show that } & \\ & \hat{P}=(r-1) /(X-1) \end{aligned}$
is an unbiased estimator for $p .[$Hint : Write out $E(\hat{P})$ as a sum, then make the substitutions
$y=x-1$ and $s=r-1 . ]$
(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.

Key Concepts

Negative Binomial Distribution

The negative binomial distribution describes the number of independent trials needed to achieve a specified number of successes. It is characterized by the probability of success on each trial and the target number of successes, with its probability mass function assigning probabilities based on the count of trials required until the final success occurs.

Unbiased Estimator

An unbiased estimator is a function of the data that, on average, equals the parameter it is intended to estimate. This means that the expected value of the estimator is exactly the true parameter value, ensuring that the estimator does not systematically overestimate or underestimate the parameter.

Expectation and Transformation Techniques

Calculating the expected value of a function of a random variable often involves summing over the probability mass function and applying algebraic substitutions or transformations to simplify the result. This process is crucial for verifying unbiasedness because it allows one to demonstrate that the expected value of the estimator equals the parameter in question.

Estimation in Sampling

Estimation in a sampling context involves using observed data to infer the value of a population parameter. By substituting observed sample values into an unbiased estimator, one can derive an estimate that, in repeated sampling, would average to the true parameter value. This forms the foundation of many inferential statistics methods.

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