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.

If $A_i$ occurs with probability $p_i$, then the expected number of $A_i$ occurring is also $p_i$. Since expected values are additive under all circumstances, $E[X]=\sum p_i$. So if this is a Poisson distribution, then necessarily one with the correct expected value, which  is $\lambda$.
