# Probability/Random Variables

## Random Variables: Definitions

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

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

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

$\{X \le x\} := \{\omega\in \Omega|X(\omega) \le x\} \in \Sigma$, 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($x_i$) = P(X = $x_i$), which has the following properties:

• p($x_i$) $\ge$ 0
• $\sum_{i} p(x_i) = 1$

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:

$P(X \in A) = \int_A f(x) dx$,

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

• $f(x)\ge\ 0\ \forall x \in\ \mathbb{R}$
• $\int_{-\infty}^{\infty} f(x) dx = 1$

### Cumulative Distribution Function

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

$F_X (x) = P(X \le x)=\begin{cases} \sum_{i: x_i \le\ x} p(x_i), & \mbox{if }X\mbox{ is discrete} \\ \, \\ \int_{-\infty}^{x} f(y) dy, & \mbox{if }X\mbox{ is continuous} \end{cases}$

The distribution function has a number of properties, including:

• $\lim_{x\to-\infty} F(x) = 0$ and $\lim_{x\to\infty} F(x) = 1$
• 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.