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Currently in a probability course and I am confused on a couple of topics: specifically the difference between them

I read the wikipedia article for a "probability space" which states that a probability space is a 3-tuple $(\Omega, \mathcal{F}, P)$ consisting of:

    1. $(\Omega)$: sample space
    1. $\mathcal{F}$: event space
    1. $P$: probability function that assigns each event in the event space a probaility between 0 and 1.

Now we are talking about random variables which are defined as "A random variable $X$ is a measurable function $X: \Omega \rightarrow S$ from the sample space $\Omega$ to another measurable space $S$ called the state space." (Wikipedia). In our class, $S$ is simply $R$: the set of real numbers. There is also the probability "distribution" defined as "a mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment". So from my understanding, the probability distribution is $\mathcal{P}: \mathcal{F} \rightarrow[0,1]$.

My questions are:

  1. Is the probability distribution function simply the third item in probability space?
  2. Or is the random variable the third item in the probability space?

I am just generally confused on how to relate the three together.

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2 Answers 2

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Short answer is you can talk about the probability distribution of a random variable which is defined as follows.

First of all random variables are measurable maps between the space $(\Omega,\mathcal{F})$ to $(\mathbb{R},\mathcal{B}_\mathbb{R})$. $\mathcal{B}_\mathbb{R}$ is the Borel sigma field. Intuitively, $\mathcal{B}_\mathbb{R}$ includes "nice" events that we care about similar to $\mathcal{F}$ but in $\mathbb{R}$. Then, the probability distribution of random variable $X$ is defined as follows. For any $A \in \mathcal{B}_\mathbb{R}$ $$ \mathrm{Pr}_X(A)=P(\{\omega \in \Omega\vert X(\omega)\in A \})\triangleq P(X^{-1}(A)). $$

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(3) is usually called the "probability function" (wiki); it seems that this function is also called the "probability distribution" (wiki).

The second link mentions different ways of describing a probability distribution.

A probability distribution can be described in various forms, such as by a probability mass function or a cumulative distribution function.

For this reason, the term "distribution function" can sometimes refer specifically to the cumulative distribution function (CDF) of a random variable which is a different function. Note that CDFs are a property of random variables, but probability functions exist on probability spaces even without the context of a random variable.

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