# PMF of two poisson random variables

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.

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.