# Probability/Random Variables

## Random Variables: Definitions

Formally, a random variable on a probability space ${\displaystyle (\Omega ,\Sigma ,P)}$ is a measurable real function X defined on ${\displaystyle \Omega }$ (the set of possible outcomes)

${\displaystyle X:\Omega \ \to \ \mathbb {R} }$,

where the property of measurability means that for all real x the set

${\displaystyle \{X\leq x\}:=\{\omega \in \Omega |X(\omega )\leq x\}\in \Sigma }$, i.e. is an event in the probability space.

### Discrete variables

If X can take a finite or countable number of different values, then we say that X is a discrete random variable and we define the mass function of X, p(${\displaystyle x_{i}}$) = P(X = ${\displaystyle x_{i}}$), which has the following properties:

• p(${\displaystyle x_{i}}$) ${\displaystyle \geq }$ 0
• ${\displaystyle \sum _{i}p(x_{i})=1}$

Any function which satisfies these properties can be a mass function.

Variables
We need some way to talk about the objects of interest. In set theory, these objects will be sets; in number theory, they will be integers; in functional analysis, they will be functions. For these objects, we will use lower-case letters: a, b, c, etc. If we need more than 26 of them, we’ll use subscripts.
Random Variable
an unknown value that may change everytime it is inspected. Thus, a random variable can be thought of as a function mapping the sample space of a random process to the real numbers. A random variable has either a associated probability distribution (discrete random variable) or a probability density function (continuous random variable).
Random Variable "X"
formally defined as a measurable function (probability space over the real numbers).
Discrete variable
takes on one of a set of specific values, each with some probability greater than zero (0). It is a finite or countable set whose probability is equal to 1.0.
Continuous variable
can be realized with any of a range of values (ie a real number, between negative infinity and positive infinity) that have a probability greater than zero (0) of occurring. Pr(X=x)=0 for all X in R. Non-zero probability is said to be finite or countably infinite.

### Continuous variables

If X can take an uncountable number of values, and X is such that for all (measurable) A:

${\displaystyle P(X\in A)=\int _{A}f(x)dx}$,

we say that X is a continuous variable. The function f is called the (probability) density of X. It satisfies:

• ${\displaystyle f(x)\geq \ 0\ \forall x\in \ \mathbb {R} }$
• ${\displaystyle \int _{-\infty }^{\infty }f(x)dx=1}$

### Cumulative Distribution Function

The (cumulative) distribution function (c.d.f.) of the r.v. X, ${\displaystyle F_{X}}$ is defined for any real number x as:

${\displaystyle F_{X}(x)=P(X\leq x)={\begin{cases}\sum _{i:x_{i}\leq \ x}p(x_{i}),&{\mbox{if }}X{\mbox{ is discrete}}\\\,\\\int _{-\infty }^{x}f(y)dy,&{\mbox{if }}X{\mbox{ is continuous}}\end{cases}}}$

The distribution function has a number of properties, including:

• ${\displaystyle \lim _{x\to -\infty }F(x)=0}$ and ${\displaystyle \lim _{x\to \infty }F(x)=1}$
• if x < y, then F(x) ≤ F(y) -- that is, F(x) is a non-decreasing function.
• F is right-continuous, meaning that F(x+h) approaches F(x) as h approaches zero from the right.