Random Variables: Definitions
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.
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.
- 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.
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
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:
- 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.