Suppose that the number of events occurring in a time period is a Poisson random variable with parameter $\lambda$. If each event is classified as a type $i$ event X
with probability $p_i$, $i = 1,… ,n, $ $\sum p_i = 1$, independently of other events, show that the numbers of type $i$ events that occur, $i = 1,… ,n$, are independent Poisson random variables with respective parameters $\lambda p_i$, $i = 1,… ,n$.
I am having trouble setting up the problem and what it's really asking asking. Any hints on how to approach this problem.
I was thinking to let $X_i$= # of events of type $i$. $X_i$~Poisson($\lambda$).
Then find $P(X_1=1,X_2=2,…,X_n=n)$. Is this the correct approach?
Best Answer
Problem setup: $X$ is Poisson with parameter $\lambda$. For each of the $X$ events, you roll an $n$-sided die and classify it as type $i$ if the roll was "$i$."
Following your notation, let $X_i$ be the number of events of type $i$. It is not true that $X_i \sim \text{Poisson}(\lambda)$ as you wrote. The question is asking you to show that $X_1, \ldots, X_n$ are independent Poisson with respective parameters $p_i \lambda$.
Hint: basically you need to show that $$P(X_1 = x_1, X_2 = x_2, \ldots, X_n = x_n) = e^{-p_1 \lambda} \frac{(p_1 \lambda)^{x_1}}{x_1!} \cdot e^{-p_2 \lambda} \frac{(p_2 \lambda)^{x_2}}{x_2!} \cdots e^{-p_n \lambda} \frac{(p_n \lambda)^{x_n}}{x_n!} \cdot$$ for any nonnegative integers $x_1, \ldots, x_n$.