Random Variables

Related Concepts

Random Variables

Given a probability space $(\Omega, \mathcal{F}, P)$, and let $\Omega$ be the sample space on which the function $X : \Omega \to \mathbb{R}$ is defined. If, for every $t \in \mathbb{R}$,

Thus, $X$ is called a random variable.

Indicator Functions

For the sample space $\Omega$, let $A$ be an event, and define the random variable

Then $I_A$ is called the indicator function of event $A$.

Distribution Functions

For a random variable $X$, the function

is called the distribution function of $X$, denoted by $X \sim F(x)$.

A distribution function has the following properties:

• Right continuity: $F(x) = F(x + 0)$
• Monotonicity: it is nondecreasing (not necessarily strictly) on $\mathbb{R}$
• $F(-\infty) = 0$,$F(+\infty) = 1$

Conversely, it can be proved that every function satisfying the requirements above is the distribution function of some random variable. Thus, distribution functions and random variables are in one-to-one correspondence.

Classification of Random Variables

According to whether its range (a random variable is a function by definition) is countable, a random variable is classified as either discrete or continuous.

Discrete Random Variables

Let $X$ be a discrete random variable with possible values $x_1, x_2, \cdots$. It can be described by a sequence of equations of the form $P\{ X = x_i \} = p_i$. This is the probability distribution learned in high school. It describes $X$.

Continuous Random Variables

Let $X$ be a continuous random variable. Studying $P\{ X = x \}$ is often meaningless, because this probability is very likely to be $0$.

On the other hand, let $X \sim F(x)$. Then

A natural idea is to use the limit $\lim\limits_{\Delta x \to 0^+} \frac{F(l + \Delta x) - F(l)}{\Delta x}$ to describe the likelihood that $X$ takes the value $l$.

This expression is the familiar derivative, so the problem becomes finding a nonnegative function $f(x)$ such that

If such an $f(x)$ exists, it is called the density function of $X$.

Independence of Random Variables

We discussed the independence of random events above. Since random variables and random events are closely related, we can similarly define independence for random variables.

Definition

If random variables $X, Y$ satisfy, for every $x, y \in \mathbb{R}$,

then $X, Y$ are called independent.

Properties

If random variables $X$,$Y$ are independent, then for any functions $f, g$, the random variables $f(X)$ and $g(Y)$ are independent.