Probability/Random Variables

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Random Variables: Definitions[edit]

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[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(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.

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:

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[edit]

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.

Independent variables[edit]