7.24* (Examination question, University of Nottingham.)

Consider a long document that consists of n(> 2) pages. It is known that one secretary started typing the document but it was completed by another.

However, exactly where in the document the second secretary took over from the first is unknown. It is proposed to model this by supposing that the first r pages were typed by the first secretary and the final n − r pages by the second where r is an unknown integer between 1 and n−1 inclusive.

The two secretaries make typing errors at different rates θ1 and θ2 per page.

Let X represent the number of errors on the ith page and assume that X has a Poisson distribution, i = 1, 2,...,n.

Then for 1 ≤ i ≤ r