Two copy editors read a $300$-page manuscript. The first found $100$ typos, the second found $120$, and their lists contain $80$ errors in common. Suppose that the author's typos follow a Poisson process with some unknown rate $\lambda$ per page, while the two copy editors catch errors with unknown probabilities of success $p_1$ and $p_2$. Let $X_0$ be the number of typos that neither found. Let $X_1$ and $X_2$ be the number of typos found only by $1$ or only by $2$, and let $X_3$ be the number of typos found by both.
(a) Find the joint distribution of ($X_0;X_1;X_2;X_3$), expressed in $\lambda$, $p_1$, $p_2$.
(b) Use the answer to (a) to find an estimates of $p_1$, $p_2$ and then the number of undiscovered typos. (Hint: Let N(s) be the Poisson process, with $T_i$ being the "time" when the i-th typo occurs, and the "time" variable is the page number.)
Will appreciate it if someone can help me with this question. I don really know how to interpret this question.
By general properties of the Poisson process, the number of errors $N$ in the manuscript follows a Poisson distribution with mean $300\lambda$. We will assume that whether an editor finds an error is independent of the other editor finding the error. Then, each error is found by neither editor, 1 only, 2 only, and both editors with probabilities $(1-p_1)(1-p_2),p_1(1-p_2),(1-p_1)p_2$ and $p_1p_2$, respectively. Thus, where $x_0+x_1+x_2+x_3=:x$,
$\Pr[(X_0,X_1,X_2,X_3)=(x_0,x_1,x_2,x_3)]=\Pr[N=x]f(x_0,x_1,x_2,x_3)$
where $\Pr[N=x]=(300\lambda)^x\exp(-300\lambda)/x!$ and
$f(x_0,x_1,x_2,x_3)={x\choose x_0,x_1,x_2,x_3}p_1^{x_1+x_3}p_2^{x_2+x_3}(1-p_1)^{x_0+x_2}(1-p_2)^{x_0+x_1}$
is the mass function of the multinomial distribution.
For (b), we are given that $X_1=20,X_2=40,X_3=80$. It is reasonable to guess that $p_1p_2=2(1-p_1)p_2=4p_1(1-p_2)$, which gives $p_2=4/5$, $p_1=2/3$. Then, $(1-p_1)(1-p_2)=1/15$, so a reasonable estimate for $X_0$ is $80/8=10$.