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I have a doubt involving the concept of random variable.

Let us consider that $\Omega = \{\omega_{1},\omega_{2},\omega_{3}\}$, $\mathcal{F} = 2^{\Omega}$ and $P(\{\omega_{i}\}) = 1/3$ for $1\leq i\leq 3$.

Having said that, let us also consider the r.v. $X:(\Omega,\mathcal{F},P)\to(\mathbb{R},\mathcal{B}(\mathbb{R}))$ defined by $X(\omega_{i}) = i$.

Then we may consider the probability of the event $A_{i} = \{\omega_{i}\}$, which is given by \begin{align*} P(A_{i}) = P(\{\omega_{i}\}) = P(X^{-1}(\{i\})) = P_{X}(\{i\}) = 1/3 \end{align*}

Here it is my question: why do we need to make use of the random variable concept if we already know the measure $P$?

After some thought, I concluded that we use random variables to convert arbitrary outcomes into numbers, which we can manipulate more comfortably. But I do not know if I am correct. Could someone please help me understand it properly?

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    $\begingroup$ You're correct. We use it to have something which we can work with comfortably. $\endgroup$ – Jakobian Jul 16 '20 at 20:25
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The point of random variables is to describe classes of events in a more elegant way than listing all of them. And in addition, since random variables are measurable functions and a lot of functional analysis also deals with measurable functions, we can use functional analysis to work with random variables. For instance, we can consider $\mathrm E[XY]$ and $\mathrm{Cov}(X,Y)$ as inner products on suitable spaces of random variables, which allows us to transfer useful stuff like the Cauchy-Schwarz inequality to the realm of probability theory. In particular, we can use everything we know about measure integrals in probability theory, since, for instance, expected values are just measure integrals w.r.t. the probability measure. So there is this whole theory we can tap into in order to describe any kind of event which can reasonably be defined via a random variable.

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Random variables are just a way to project an abstract space onto something we can work with more easily. That is very useful if we want to have many different random variables, but we do not really care about the space. That gives us the possibility to say we have some (possibly weird and complicated) abstract probability space $(\Omega, , \mathcal{F}, \mathbb{P})$ and many (e.g. independent) random variables $X,Y,...$ for example with $X$ uniform on $[0,1]$ and $Y$ exponentially distributed on $[0,+\infty)$. This gives us even the possibility to define uncountably many random variables on the same abstract space.

Very often, you will see that a probabilistic paper starts with "All random variables are defined on the abstract space $(\Omega, , \mathcal{F}, \mathbb{P})$" without being concerned how it looks like. That is the great advantage of random variables.

On the other hand, something I found very confusing at the beginning, you sometimes look at random variables which look very stupid at first sight. They are sometimes called the canonical rv and are used if the space $\Omega$ is defined explicitly. For example, if you look at random functions $\Omega = C[0,1]$, then you might define the random variable $X_t: \omega \mapsto \omega(t)$. With the right probability measure on $C[0,1]$, you then get a random function $X$ on $[0,1]$ which is continuous for fixed $\omega\in\Omega$.

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Speaking in layman's terms, rvs allow us to 'hack' into the sample space $\Omega$ by deriving its subsets/pre-images/events $A_i$. Once preimage is derived, its probability/Lebesgue measure is easy to find. To this extent you need to have the map $X:\Omega \to \mathbb{R}$ ('random' 'variable') 1) defined in some functional way and 2) measurability, because event $A_i$ is a subset of $\Omega$ and an element of $\mathcal{F}$. So rvs are an essential link between sample space and sigma-algebra.

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