i've seen in some tutorials and courses that the poisson process formula is :
$P(X=n) = e^\left(-\lambda\right) \frac{\lambda ^n}{n!}$
but facing some problems using this formula i did some researches and i found elsewhere that the Poisson process formula is :
$P(X(t)=n) = e^\left(-\lambda t\right) \frac{(\lambda t) ^n}{n!}$
i'm confused which one to use and when, i had a problem saying :
an electronic component follow an exponential distribution with parameter lambda $\lambda = 1 h^{-1}$ . Once any device fails, it is immediately replaced by a identical one.
Q : What is the probability of having more than three failures within a range of two hours?
first I have not thought about using poisson formula, but after some researches i found out that it is the more convenient solution. i couldn't use the first formula since $\lambda = 1 h^{-1}$ and the question is about an interval of 2h.
i found a solution that use the second formula like this : $P(X(2)\ge 3) = 1 - P(X(2) < 3) = .... = 0.1429$
i understand how they used it and i even calculated it and found the right result, but how can i know that i have to use the second formula and is it possible to use in any poisson process ?
thank you very much in advance.
An essential distinction between Poisson distributions and Poisson processes is the introduction of a parameter "t" (for time) or "a" (for area); it's like the introduction of a supplementary dimension (or, said otherwise, as the introduction of units, as a physicist would consider it).
But there is another essential characteristic (or better said, axiom) of the Poisson process that has no equivalent in the Poisson distribution and which makes all the difference: the independence of events happening in disjoint time (or space) intervals. This is called the second key property in (https://en.wikipedia.org/wiki/Poisson_point_process). The introduction of this second property, well explained in the Wikipedia article has slowly been introduced 100 years after the discovery of the Poisson distribution, which means that it wasn't evident "per se".
Edit: here is an example of application of the "2nd key" property.
Consider a space interval $[0,a]$ (that could be as well thought of as a time interval). Let $b$ with $0<b<a$: for example $a=1$ and $b=0.6.$
$\circ--------------\circ------\circ$
$0-------------- \ b------ \ a$
Consider a random variable $X$ uniformly distributed between $0$ and $a$.
Let us consider $n$ realizations of $X$.
Let $Y$ be the number of these realizations that fall into $[0,b]$.
Question: what is $P(Y=k)$ for $k=0,1,2,\cdots n$ ?
First method: $Y$ has a binomial $B(n,p)$ distribution with $p=\frac{b}{a}$ (ratio of lengths); thus
$$\tag{1}P(Y=k)=\binom{n}{k}p^k(1-p)^{n-k}$$
Second method (using the Poisson process): Consider the equivalent conditional probability
$$P(k \ \text{hits in} \ [0,b] \ | \ n \ \text{hits in} \ [0,a])$$
$$=\dfrac{P(k \ \text{hits in} \ [0,b] \cap n \ \text{hits in} \ [0,a])}{P(n \ \text{hits in} \ [0,a])}$$
$$=\dfrac{P(k \ \text{hits in} \ [0,b] \ \cap \ (n-k) \ \text{hits in} \ [b,a])}{P(n \ \text{hits in} \ [0,a])}$$
Which, because of the "2nd key property", can be transformed into:
$$=\dfrac{P(k \ \text{hits in} \ [0,b]) \times P((n-k) \ \text{hits in} \ [b,a])}{P(n \ \text{hits in} \ [0,a])}$$
$$=\dfrac{\dfrac{e^{-bt}(bt)^k}{k!}\dfrac{e^{-(a-b)t}((a-b)t)^{n-k}}{(n-k)!}}{\dfrac{e^{-at}(at)^n}{n!}}$$
We can operate big simplifications, yielding:
$$\tag{2}\dfrac{n!}{k!(n-k)!}\left(\dfrac{b}{a}\right)^k\left(\dfrac{a-b}{a}\right)^{n-k}.$$
(1) is exactly retrieved!
What is the interest of the second method ? Not its efficiency (the first method provided an almost instant answer). Its merit is:
to make it evident that, thanks to the Poisson process concept, Poisson distribution is the "master of the game", from which Binomial distribution can be derived.
to give a first "technical" idea of the way this "2nd key property" can be used.
Remark: This "transformation" Poisson $\rightarrow$ Binomial is not to be confused with the Poisson limit theorem (which is an approximation, valid under certain conditions, of a Binomial distribution by a Poisson distribution (https://en.wikipedia.org/wiki/Poisson_limit_theorem)).