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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.

Could you please help me with this problem ? Thank you very much!

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  • $\begingroup$ 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. $\endgroup$
    – John Douma
    Commented May 15, 2021 at 14:20
  • $\begingroup$ 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. $\endgroup$
    – drhab
    Commented May 15, 2021 at 14:22

1 Answer 1

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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 $S$.

All this machinery is fairly easy for discrete distributions, where things get complicated are for continuous distributions.

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  • $\begingroup$ 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! $\endgroup$ Commented May 15, 2021 at 14:27
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    $\begingroup$ @InTheSearchForKnowledge, my knowledge here is shaky, but a "measure" is essentially a formula that assigns lengths or areas or volumes to subsets. So if you want to use length as probability, I think that would imply that the distribution has to be uniform. $\endgroup$ Commented May 15, 2021 at 14:30
  • $\begingroup$ @RoberTheTutor: Hi, so, for example, in the case of Normal distribution, do we use the Lebesgue measure as the Probability measure or not please ? Thank you very much for your help! $\endgroup$ Commented May 15, 2021 at 14:32
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    $\begingroup$ @InTheSearchForKnowledge, again, I'm not an expert on the part, but reading a few Wikipedia articles leads me to believe that no, the only distribution that uses the Lebesgue measure as the probability measure is the uniform distribution. You are asking when probability is proportional to length; that implies the probability density is a constant. $\endgroup$ Commented May 15, 2021 at 14:38

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