Importance of a Random Variable [duplicate]

Question

I've been reading about random variables and I don't get as to what is the importance of a random variable in statistics? I mean why is it a useful concept?

Answer 1

Given a sample space of all outcomes of a probabilistic experiment $\Omega =\{\omega_1,\omega_2,...,\omega_n\}$. The random variable $\eta$ is a mapping from sample space to set of real numbers i.e. $$\eta:\Omega\mapsto\mathbb{R}$$

So essentially random variables give some numeric characteristic of an outomce. For the experiment of rolling two dice the outcomes of each die are from $S = \{1,2,...,6\}$ and the sample space is $\Omega = S \times S = \{(1,1), (1,2),...,(6,6)\}$; so we may define a random variable that represents the sum of 2 dice i.e. $\eta((1,1)) = 2$. Therefore, if the value of interest is the sum of the 2 dice then we encapsulate the necessary numerical values with random variables; thus, having a compact notation for computing desired probabilities. Moreover, the mapping enables to define cumulative distribution function (c.d.f) of random variable i.e.

$$F_\eta(x) = \mathbb{P}(\omega:\eta(\omega) \leq x)$$

So rather than working with sample spaces and events (for example, for computing the probability of having the sum of 2 dice less than 4), we can compute probabilities using the c.d.f as a shorthand notation for desired probability.