Let (X) have the negative binomial distribution (f(x)=left(begin{array}{l}x-1 r-1end{array} ight) p^{r}(1-p)^{x-r}) for (x=r, r+1, ldots) (a)

Let \(X\) have the negative binomial distribution

\(f(x)=\left(\begin{array}{l}x-1 \\ r-1\end{array}\right) p^{r}(1-p)^{x-r}\) for \(x=r, r+1, \ldots\)

(a) Obtain the maximum likelihood estimator of \(p\).

(b) For one engineering application, it is best to use components with a superior finish. Suppose \(X=\) 27 identical components are inspected, one at a time, before the \(r=3 \mathrm{rd}\) component with superior finish is found. Find the maximum likelihood estimate of the probability that a component will have a superior finish.