"A first Course in probability"-chaper4-discrete random variable-poisson distribution

Based on the concept of instrumental characterization of probabilistic problems, such as random variables, distribution columns, we can begin to discuss a variety of distribution columns. (This chapter is called "Random variables" in the book, but in order to separate from chapter fifth "Continuous random variables", here the title is "discrete Random variable")

Poisson distribution derived from the two-item distribution combined series:

We are familiar with the two distributions and we are very popular in our life, but the calculation formula is a bit cumbersome, we now make the following simplification deduction:

Set the parameter for a two-item distribution (N,P), setting the parameter λ=np. The random variable is x.

At the same time, combining several limit methods, we can see that when n approaches infinity, there are:

So we get:

This is the Poisson distribution column. It is easy to see that the two distributions of n approaching Infinity can be equivalent to the Poisson distribution, and we can verify the Poisson distribution as a property of the distribution column if it is based on n approaching Infinity:

Ps: The derivation process uses the Taylor series expansion, the specific content of the author in the "Thomas University Calculus" column will be given.

"A first Course in probability"-chaper4-discrete random variable-poisson distribution