Application of Poisson probability to quantify fractions of the population

Question

Let's say that $X$ is a r.v. which obeys Poisson distribution with a given $\lambda$, and represents the number of events occurring in a set of elements. We can easily calculate the value of $P(X = x) = f(x) = \hat{p}$.

Now let's say that our set has $n$ elements. Someone uses to say that $n \times \hat{p}$ of them will experience $x$ events. It does make sense but, of course, things can go also differently.

Does $n \times \hat{p}$ represent a kind of expected values of elements hit by $x$ events? I do not understand the limit of those kind of direct "quantifications".

PS: I hope the title of this post is sufficiently clear :)

Answer 1

You can use this to investigate if n real elements are (approximately) poisson distributed.

Lets say you have a list of number of elements which are occuring in one time unit. $x$ is the number of elements which are occuring in one time unit. And you have make this experiment 12169 times. The results are listed in the second column. You think that x is poisson distrubuted. By using the theoretical distribution multiplied by n you can see if your conjecture is is more or less right.

x       n          P(X=x)* n
0       5267       5268.7
1       4436       4410.5
2       1800       1846.0
3        543        515.1
4        111        107.8
5         21         20.9
...
sum n    12169

$\overline x=\frac{4436+2*1800+3*534+4*111+5*21}{12169}=0.8371$

$P(X=x)\cdot n= e^{-0.8371}\cdot \frac{0.8371^x}{x!}\cdot 12169$

Comparing the two colums we can assume that the elements are poisson distributed.