Poisson paradigm: Why is $\lambda$, the rate of occurrence of events, equal to the sum of the probabilities of all the events that occur? My notes say the following about the Poisson paradigm:

Let $A_1, A_2, \dots, A_n$ be events with $p_j = P(A_j)$, where $n$ is large, the $p_j$ are small, and the $A_j$ are independent or weakly dependent. Let
$$X = \sum_{j = 1}^n I(A_j)$$
count how many of the $A_j$ occur. Then $X$ is approximately $Pois(\lambda)$ with $\lambda = \sum_{j = 1}^n p_j$.

$\lambda = \sum_{j = 1}^n p_j$ is the sum of all of the probabilities of the events that occur. And $\lambda$ is the rate of occurrence of events. But I'm wondering why $\lambda$, the rate of occurrence of events, would be equal to the sum of the probabilities of all the events that occur? I'm not seeing how this makes sense.
I would greatly appreciate it if people could please take the time to clarify this.

In the independent case, it is basically like the Poisson approximation to the binomial, which you can see by a direct calculation of a limit. In the weakly dependent case, you're basically approximating by pretending the dependence isn't there and then applying the previous form.
A more abstract way of viewing it is that the expected number of occurrences is definitely $\sum_{i=1}^n p_i$ regardless of any assumptions about the dependence, so whatever approximation that you make should at least have the same expected number of occurrences as the actual expected number. Since the Poisson only has one parameter anyway, this criterion locks down which Poisson you can use.
