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.