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In a french book, "Calcul des probabilités" from Foata and Fuchs, I found this theorem, which they call "Poissonization".

"Let $(I_k)_{k \in \mathbb{N}}$ be a sequence of independent variables with Bernoulli distribution, each of the same parameter, $p$. Let $N$ be a variable integer-valued independent from the $I_k$, and define $N_1:= \sum_{k=1\ldots N} I_k$, and $N_2:=\sum_{k=1\ldots N} (1−I_k)$.

Then if $N$ has a Poisson distribution with parameter $\lambda$, then $N_1$ and $N_2$ are independent, and have Poisson laws, of parameters $\lambda p$ and $\lambda (1-p)$.

Conversely, if $N_1$ and $N_2$ are independent, then $N$ has a Poisson distribution."

What does this situation modelize ? Is there a natural problem where this situation arises ? I really don't understand what I am dealing with.

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Suppose that $N$ represents the number of customers who have arrived to a store up to time $t$, and that $I_k$ is an indication of whether the $k^\text{th}$ customer is male. Then $N_1$ counts up the number of male customers and $N_2$ counts up the number of female customers.

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