Intuition concerning this characterization of the Poisson law

Question

I found the following characterization of the Poisson law in a French book about probabilities.

Let $N$ be a random variable with values in $\mathbb N$. Let us flip a fair coin $N$ times successively and independantly of the previous tosses, with probability $0.5$ of getting any side of the coin. Let $X$ be the number of heads we obtain, and $Y$ be the number of tails. Then $X$ and $Y$ are independant if and only if $N$ follows a Poisson law.

While I can easily understand the proof and the computations which are going on, I fail to understand the intuition behind it: how could we have expected the Poisson law to be the one ?

Thank you very much for your help.

Answer 1

One reason why it would be natural to suspect Poisson distribution is the following property:

If $X_1\sim\operatorname{Poisson}(\lambda_1)$ and $X_2\sim\operatorname{Poisson}(\lambda_2)$ are independent, then $X_1+X_2\sim\operatorname{Poisson}(\lambda_1+\lambda_2)$ and $X_1\mid(X_1+X_2)\sim\operatorname{Binomial}(X_1+X_2,\frac{\lambda_1}{\lambda_1+\lambda_2})$.

Now it is clear that in our case $X\mid(X+Y)\sim\operatorname{Binomial}(X+Y,\frac12)$, so we might hope for $X,Y$ being independent Poissons with the same rate. Then the computation shows indeed that is the case.