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.


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.

  • $\begingroup$ 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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