Probability/Random Variables

Index

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Random Variables: Definitions

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

$X:\mathrm{\Omega }\to ℝ$,

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

$\{X\le x\}:=\{\omega \in \mathrm{\Omega }|X(\omega )\le x\}\in \mathrm{\Sigma }$, 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.

Continuous variables

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

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

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

• $f(x)\ge 0\mathrm{\forall }x\in ℝ$
• $\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}f(x)dx=1$

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:

$F_X(x)=P(X\le x)=\{\begin{array}{cc}\sum _{i:x_i\le x}p(x_i),& \text{if }X\text{ is discrete}\\ \\ \underset{-\mathrm{\infty }}{\overset{x}{\int }}f(y)dy,& \text{if }X\text{ is continuous}\end{array}$

The distribution function has a number of properties, including:

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