# Real life example of set-theoretic definition of probability space and random variable

I'am new to probability theory and I have some difficulties in relating the definition of a probability space and random variable with the real life.

Definition of probability space:
Suppose $$\mathcal{F}$$ is a $$\sigma$$ - algebra on the set $$\Omega$$ and $$\mathbb{P}$$ is a probability measure on $$(\mathcal{F}, \Omega)$$ then the triple $$(\mathcal{F}, \Omega, \mathbb{P})$$ is called a probability space

Definition of a random variable:
A random variable $$X$$ on the probability space $$(\mathcal{F}, \Omega, \mathbb{P})$$ is a measurable function from $$\Omega \to \mathbb{R}$$

When I come up with an example like:

Suppose $$X$$ is a random variable which denotes the number of traffic accident in the country A in $$1$$ year.
Suppose $$X$$ follows the Poisson distribution with parameter $$\lambda$$

I can't figure out which is $$\Omega$$, which is $$\mathcal{F}$$, the probability measure $$\mathbb{P}$$ and the function $$X$$ (I mean the expression of $$X$$) in this example of Poisson distribution.

• Omega is the set of outcomes of an experiment, $\mathcal F$ is the set of events and $\mathbb P$ is the measure that assigns a probability to each of the events. A random variable assigns a number to each of the outcomes. See en.wikipedia.org/wiki/Probability_space for some examples. Commented May 15, 2021 at 14:20
• If you only need some rv $X$ having Poisson distribution with parameter $\lambda$ then you can go for: $\Omega=\mathbb N$, $\mathcal F=\mathcal P(\mathbb N)$, probability measure $\mathbb P$ is is prescribed by $A\mapsto\sum_{n\in A}p(n)$ where $p(n)=e^{-\lambda}\frac{\lambda^n}{n!}$ and $X:\mathbb N\to\mathbb R$ is prescribed by $n\mapsto n$. Do not think though that this is the only possibility. Commented May 15, 2021 at 14:22

The sample space $$\Omega$$ is the set of all possible outcomes. So for a Poisson process, $$\Omega = \{0,1,2,3,...\}$$.
The $$\sigma$$-algebra is the set of all collections of possibilities that have probabilities. In other words, the set of all subsets of $$\{0,1,2,3...\}$$
The $$\mathbb{P}$$ is the formula that assigns probabilities to every element of $$\Omega$$, and also to any subset $$S$$ of $$\Omega$$ (in this instance by simply summing the probabilities of all the possibilities in $$S$$.
• Thank you very much for your response. Concerning the $\mathbb{P}$, in this case, is it Lebesgue measure ? If it is not then could you please name another distribution that use the Lebesgue measure as the probability measure ? I'm quite confuse when to use Lebesgue measure, when to use another measure. Thanks for your help! Commented May 15, 2021 at 14:27