Question: Suppose that the number of typographical errors in a new text is Poisson distributed with mean \(\lambda\). Two proofreaders independently read the text. Suppose that each error is independently found by proofreader \(i\) with probability \(p_{i}, i=1,2\)

Let \(X_{1}\) denote the number of errors that are found by proofreader 1 but not by proofreader 2. Let \(X_{2}\) denote the number of errors that are found by proofreader but not by proofreader 1 . Let \(X_{3}\) denote the number of errors that are found by both proofreaders. Finally, let \(X_{4}\) denote the number of errors found by neither

proofreader

1. Describe the joint probability distribution of \(X_{1}, X_{2}, X_{3}, X_{4}\).
2. Show that
   \[\frac{E\left[X_{1}\right]}{E\left[X_{3}\right]}=\frac{1-p_{2}}{p_{2}} \text { and } \frac{E\left[X_{2}\right]}{E\left[X_{3}\right]}=\frac{1-p_{1}}{p_{1}}\]
   Suppose now that \(\lambda, p_{1}\), and \(p_{2}\) are all unknown.
3. By using \(X_{i}\) as an estimator of \(E\left[X_{i}\right], i=1,2,3\), present estimators of \(p_{1}\), \(p_{2}\), and \(\lambda\).
4. Give an estimator of \(X_{4}\), the number of errors not found by either proof-

reader.

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