I am stuck on a problem which actually I have the answer to. First, let me give you another example question then I'll try to explain why I am stuck on this other question.
The emails I get can be modeled by a Poisson distribution with an 0.2 emails per minute. Then what is the probability that I get no emails in an interval of 5 minutes?
The solution is as follows:
Since I get 0.2 emails per minute, $\lambda = 0.2 \times 5 =1$. Then $P(X=0)=\frac{e^{-\lambda} \lambda^k}{k!} = \frac{1}{e}$
Okay now let's get to the question I got stuck.
Let $X\sim \text{Poisson}(\alpha)$ and $Y\sim \text{Poisson}(\beta)$ be two independent random variables. Define a new random variable as $Z = X+Y$. Find the PMF of $Z$.
The problem is solved as follows:
$$P_Z(k) = P(X+Y =k)$$ $$= \sum_{i = 0}^k P(X+Y=k\mid X=i)P(X=i)$$ $$= \sum_{i = 0}^k P(Y=k-i)P(X=i)$$ then from the Binomial theorem it ends up being: $$= \frac{e^{-(\alpha + \beta)}}{k!}(\alpha + \beta)^k$$
I have no problems following the solution and understanding why the first statement ends to the last one. What I don't understand is how can we just write $P_Z(k) = P(X+Y =k)$. In the first example to calculate $\lambda$, we multipled 0.2 with 5 since we receive 0.2 emails per minute and we are asked about a 5 minute interval. What if this $\alpha$ and $\beta$ parameters belong to different time framed distributions? If $\alpha$ is for per minute and $\beta$ is for per hour don't we have to do something like $P_Z(k) = P(60X+Y =k)$?
Thank you