§ Poisson distribution

• Think about flipping a biased coin with some bias $p$ to associate a coin flip to each real number. Call this $b: \mathbb R \to \{0, 1\}$.
• Define the count of an interval $I$ as $\#I \equiv \{ r \in I | b(r) = 1 \}$.
• Suppose that this value $\#I$ is finite for any bounded interval.
• Then the process we have is a poisson process.
• Since the coin flips are independent, all 'hits' of the event must be independent.
• Since there is either a coin flip or there is not, at most one 'hit' of the event can happen at any moment in time.
• Since the bias of the coin is fixed, the rate at which we see $1$s is overall constant.