36. Suppose that the number of typographical errors in a new text is Poisson distributed with mean A. Two proofreaders independently read the text. Suppose that each error is independently found by proofreader / with probability pi9 i = 1, 2. Let Xx denote the number of errors that are found by proofreader 1 but not by proofreader 2. Let X2 denote the number of errors that are found by proofreader 2 but not by proofreader 1. Let X3 denote the number of errors that are found by both proofreaders. Finally, let X4 denote the number of errors found by neither proofreader.

(a) Describe the joint probability distribution of Xl9 X2, X3, X4.

(b) Show that E[XX] = l - p 2 d E[X2] 1 - A E[X3] p2 an E[X3] Pl Suppose now that A, px, and p2 are all unknown.

(c) By using X{ as an estimator of E[Xt]9 i = 1, 2, 3, present estimators of Piy Pa a nd A.

(d) Give an estimator of X4, the number of errors not found by either proofreader.