# Suppose that the number of typographical errors in a new text is Poisson distributed with mean 1 answer below »

Suppose that the number of typographical errors in a new text is Poisson distributed with mean λ. 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 2 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.

(a) Describe the joint probability distribution of X_{1},X_{2},X_{3},X_{4}.

(b) Show that

Suppose now that λ, p_{1}, and p_{2} are all unknown.

(c) By using X_{i} as an estimator of E[X_{i} ], i = 1, 2, 3, present estimators of p_{1}, p_{2}, and λ.

(d) Give an estimator of X_{4}, the number of errors not found by either proofreader.