Undergrad Proof to the Expression of Poisson Distribution

[Anakin Skywalker]

Hello.
Given a range of time in which an event can occur an indefinite number of times, we say a random variable X folows a poisson distribution when it follows this statements:

• X is the number of times an event occurs in an interval and X can take values 0, 1, 2, …
• The occurrence of one event does not affect the probability that a second event will occur. That is, events occur independently.
• The rate at which events occur is constant. The rate cannot be higher in some intervals and lower in other intervals.
• Two events cannot occur at exactly the same instant.
• The probability of an event in an interval is proportional to the length of the interval.

And in this case the probability related to x is given by the expression below:

I would like to know how this expression is deduced.

P.S.: I used the informations in the wikipedia's page, so I'm not so sure that these topics are right.
https://en.wikipedia.org/wiki/Poiss..._Poisson_distribution_an_appropriate_model.3F

 

[andrewkirk]

The Poisson distribution can be deduced as a limiting case of the binomial distribution.

Let the probability of exactly one event occurring in a period of length $\delta t$ be $g(\delta t)$, and define
$$\lambda\triangleq \lim_{\delta t\to 0}\frac{g(\delta t)}{\delta t}$$

Then if $X(T)$ is the number of events occurring in a period of length $T$ and $n$ is a positive integer, we have
$$Pr(X(T)\leq x)=Pr (B_n\leq x)$$
where $B_n$ is a binomial random variable with parameters $(n,g(T/n))$.

Hence $Pr (B_n\leq x)$ is constant over $n$ and hence its limit as $n\to\infty$ exists and is equal to $Pr(X(T)\leq x)$.

But the number of events occurring in a period of length $T$ is also equal to the Poisson random variable $Y(\lambda,T)$ for the number of events in time period $T$ with frequency $\lambda$.

So we have
$$Pr(Y(\lambda,T)\leq x)=\lim_{n\to\infty}Pr(B_n\leq x)$$
and it can be shown that the expression on the RHS is equal to the formula for the Poisson CDF.

I've left some steps out because I'm a bit rushed this morning, but this should at least give an idea of how it's done.