Random Variables and Distributions - Core Definitions

Random variables provide the bridge between abstract probability spaces and numerical analysis. They transform outcomes into numbers, enabling us to perform calculations and comparisons.

Random Variables

Definition

A random variable is a function $X: \Omega \to \mathbb{R}$ that assigns a real number to each outcome in the sample space. Formally, $X$ must be measurable: for every real number $x$ , the set $\{\omega \in \Omega: X(\omega) \leq x\}$ must be an event (i.e., belong to $\mathcal{F}$ ).

Random variables are denoted by capital letters ( $X, Y, Z$ ) while their values are denoted by lowercase letters ( $x, y, z$ ).

Example

Coin Tosses: Toss a fair coin three times. Define $X$ = number of heads.

Sample space: $\Omega = \{HHH, HHT, HTH, HTT, THH, THT, TTH, TTT\}$
$X(HHH) = 3, X(HHT) = 2, X(HTH) = 2, \ldots, X(TTT) = 0$
$X$ takes values in $\{0, 1, 2, 3\}$

Cumulative Distribution Function

Definition

The cumulative distribution function (CDF) of a random variable $X$ is: $F_X(x) = P(X \leq x) = P(\{\omega \in \Omega: X(\omega) \leq x\})$

for all $x \in \mathbb{R}$ .

The CDF completely characterizes the distribution of $X$ . It satisfies three properties:

Monotonicity: $F_X$ is non-decreasing
Limits: $\lim_{x \to -\infty} F_X(x) = 0$ and $\lim_{x \to \infty} F_X(x) = 1$
Right-continuity: $\lim_{h \to 0^+} F_X(x+h) = F_X(x)$

From the CDF, we can compute probabilities of intervals: $P(a < X \leq b) = F_X(b) - F_X(a)$

Discrete vs. Continuous Random Variables

Definition

A random variable $X$ is discrete if it takes on countably many values. Its distribution is characterized by a probability mass function (PMF): $p_X(x) = P(X = x)$

A random variable $X$ is continuous if its CDF is continuous and differentiable almost everywhere. Its distribution is characterized by a probability density function (PDF): $f_X(x) = \frac{d}{dx} F_X(x)$

satisfying: $P(a \leq X \leq b) = \int_a^b f_X(x) \, dx$

For continuous random variables, $P(X = x) = 0$ for all $x$ , so we focus on probabilities of intervals.

Example

Discrete: Rolling a fair die, $X \in \{1,2,3,4,5,6\}$ with $p_X(k) = 1/6$ for each $k$ .

Continuous: Uniform distribution on $[0,1]$ has PDF $f_X(x) = 1$ for $0 \leq x \leq 1$ (and 0 elsewhere).

Remark

The distinction between discrete and continuous random variables parallels the distinction between counting and measuring. Discrete variables count outcomes (number of successes, number of arrivals), while continuous variables measure quantities (time, weight, distance).