Random Variable

Topics: Probability

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(definition)

A random variable $X$ is a measurable function $X:\Omega \to \mathbb{R}$ ($\Omega$ a sample space) where, given any set $B$:

$$X^{-1}(B)\in \mathcal{F}$$

…or also:

$$\left\{w\in \Omega \mid X(w)\le x\right\}\in \mathcal{F}$$

…where $\mathcal{F}$ is a sigma-algebra.

In other words, a random variable is a mapping from the possible outcomes in the sample space to a measurable space (in this case $\mathbb{R}$).

As such, in general, we can see a random variable as the mathematical formalisation of quantities or objects that depend on randomness.

Notation

Uppercase letters, like $X,Y,Z$, denote the random variable itself. On the other hand, lowercase letters, like $x,y,z$, denote a value that the corresponding random variable may take.

With this, we can write the range of a random variable $X$ as:

$$R_X=\{x_1,x_2,x_3,\dots \}$$

Types §

There are two types of random variables:

• Discrete random variables
• Absolutely continuous random variables

For both of these, we can define a density function, which helps us specify the probability of the random variable being a certain value or being in a given interval.