Chapter VI · 18 min

Random Variables

Numbers attached to outcomes

“A random variable is a function, and as such it is neither random nor a variable.”

Outcomes can be messy — a whole deck, a whole image, a whole game of Monopoly. What you can compute with is often a number you attach to that mess: the number of aces, the pixel intensity, the total score. A random variable is that attachment. It is a function, and as such it is neither random nor a variable. The randomness lives on $\Omega$; the function just reports a number.

A function on Ω

A (real) random variable is a map $X:\Omega \to \mathbb{R}$. For a subset of the line, the event that $X$ lands there is the preimage

We write $P(X\le x)$ as shorthand for $P(\{\omega :X(\omega )\le x\})$. The left side looks like a statement about a number; the right side is an ordinary event. Keeping both pictures in mind prevents a surprising amount of confusion.

Distributions, pmfs, cdfs

The distribution of $X$ is the push-forward of $P$ onto the line: the rule that tells you $P(X\in B)$ for sets of numbers. When $X$ takes finitely or countably many values, the distribution is recorded by a probability mass function

The cumulative distribution function (cdf)

is defined for every random variable, discrete or not. It is non-decreasing, right-continuous, and runs from 0 to 1. For a discrete variable it is a staircase; the height of the jump at $x$ is $p_X(x)$.

• Two different experiments can induce the same distribution. Fair-coin $0/1$ and “I drew an even die face” are different stories with the same $p$.
• Functions of random variables are random variables: $Y=g(X)$ has $Y(\omega )=g(X(\omega ))$.
• The pair $(X,Y)$ is a random vector — a function to ${\mathbb{R}}^2$. Chapter 10.