PROBABILITIES Course

Poisson distribution

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It's called the loi des événements/phénomènes rares in French. The Poisson distribution $\mathbb{P}(\lambda)$ require that the probability $p$ is relatively smaller than $n$, while the parameter $\lambda$ is equals to $n*p$.

  • The mass function is $\mathbb{P}(X=k) = \frac{\lambda^k * e^{-\lambda}}{k!}$
  • $E(X) = \lambda$
  • $V(X) = \lambda$

Demonstration

You need to understand this demonstration because you are going to do something like this a lot, and it's easy

Demonstration of $E[X]=\lambda$

With the expected value formula, we have

  • $\sum_{k \in \mathbb{N}} k * P(X=k)$
  • $= \sum_{k \in \mathbb{N}} k * \frac{\lambda^k * e^{-\lambda}}{k!}$
  • $e$ is a constant, so we can extract it
  • $= e^{-\lambda} * \sum_{k \in \mathbb{N}} \frac{\lambda^k}{(k-1)!}$
  • I'm taking one $\lambda$ out (it's a constant too)
  • $= \lambda e^{-\lambda} * \sum_{k \in \mathbb{N}} \frac{\lambda^{k-1}}{(k-1)!}$

Here we are starting the magic. There are well-known series that can be replaced by a function.

Check them here, but this is the only one that we will use: $\sum_{k \in \mathbb{N}} \frac{\lambda^{k}}{k!} = e^k$

  • so using the exponential series development
  • $= \lambda e^{-\lambda} * e^{\lambda}$
  • $= \lambda e^{-\lambda+\lambda}$
  • $= \lambda e^{0}$
  • $= \lambda * 1$
  • $= \lambda$

Remember the magic trick, it's useful.