# Poisson Process

I got interested in the Poisson Process. Poisson Process is beautiful.

This post is dedicated to the connection between the Bernoulli Trials and the Poisson Process along with the Generalizations of the Poisson Process.

#### Resources:

- This describes the relationship between the Bernoulli Trials and Poisson Process. ( Extension of Bernoulli Process ).
- Bernoulli Process is a sequence of iid Bernoulli Trials.

- A Poisson Process can’t be extended to the Higher Dimensions w.r.t the time perspective as a Stochastic Process due to the absence of order in the higher dimensional space. So, it can’t be extended as a Stochastic Process, rather it can be done as a Counting Process. For that one can refer to the initial pdf file given!
- Another important thing that caught my eyes is that if \( X ~ Poi (a), Y ~ Poi (b) \) then \( X | Y ~ Bin ( Y, \frac{a}{b} ) \). This idea can be generalized to Poisson Process in higher dimension in problem 14 of the first link.
- This is also the first example of Continuous-Time Markov Chain.