# Probability/Random Variables

## Contents

## Random Variables: Definitions[edit]

Formally, a *random variable* on a probability space is a measurable real function *X* defined on (the set of possible outcomes)

- ,

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

- , i.e. is an event in the probability space.

### Discrete variables[edit]

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() = P(X = ), which has the following properties:

- p() 0

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[edit]

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

- ,

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

### Cumulative Distribution Function[edit]

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

The distribution function has a number of properties, including:

- and
- 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.