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A book publisher employs two typists, X and Y. Typist X makes typographical errors at the rate of k per page, and Y makes them at a different rate of r per page.
a. Considering that both X and Y do half of the entire publisher's typing, write down an expression for the PMF of the random variable E, the number of errors on a randomly chosen page.
b. Write down the PMF of E if the typist with the error rate k types 70 percent of the pages.

According to merging principle, my PMF should be distributed over Poisson(k+r) but I'm confused with the "half of the entire publisher's typing" and "70% of pages". I'm unable to incorporate this information. Any help would be appreciated.

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The merging principle is only when you are counting the total number of Poisson distributed events in the same time interval. But each page is a separate interval.

It is even easier: By the law of total probability, \begin{align} P(E = e) &= P(E = e \mid \text{$X$ typed the page}) P(\text{$X$ typed the page}) \\ &\qquad\qquad+ P(E = e \mid \text{$Y$ typed the page}) P(\text{$Y$ typed the page}). \end{align} The first factor of each term is the Poisson pmf corresponding to the error rate of the typist.

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  • $\begingroup$ Thanks. That makes sense. But we are assuming here that each page is typed by only one typist. What if that's not the case? $\endgroup$ – user2517839 Sep 30 '16 at 22:31
  • $\begingroup$ You are assuming Poisson typists. But the problem did not specify the distribution of errors per page attained by each typist. For all we know, typist Y van be perfect but spiteful, and commit precisely 13 errors on every page she types. $\endgroup$ – Mark Fischler Sep 30 '16 at 22:35
  • $\begingroup$ Got it. And if I have to calculate the expected value of this PMF, would it be sum of corresponding weighted means? $\endgroup$ – user2517839 Sep 30 '16 at 23:01
  • $\begingroup$ Yeah (that is the law of total expectation, which is easily derived from what you have). $\endgroup$ – arkeet Sep 30 '16 at 23:05

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