The number of mistakes that are made by a writer on each page is a Poisson distributed random variable with parameter $\lambda$, independent for each page. A 3-page article is being rewritten by one of 4 writers, for who $\lambda$'s are equal respectively to $1$, $2$, $3$, $4$. Writer is chosen randomly with equal probability. What is the expected value of mistakes in whole article?
My attempt $X$ - number of mistakes made on single page $$\mathbb E(X)=1\cdot\frac{1}{4}+2\cdot\frac{1}{4}+3\cdot\frac{1}{4}+4\cdot\frac{1}{4}=2.5$$
$$3\cdot2.5=7.5$$
Anwser: $7.5$
Is that correct?
Yes, you are using the conditional expectation formula over the writers $W_i$ to get 2.5, $$E(X) = \sum E(X|W_i)P(W_i) $$
and then linearity of expectation to get $3\times2.5=7.5$ $$E(nX) = nE(X)$$