Cumulative Distribution Functions

Definition

The cumulative distribution function (CDF) of a random variable $X$ is:

$F(x) = P(X \leq x)$

It gives the probability that $X$ is less than or equal to $x$.

Why CDFs?

The CDF has advantages over the PMF/PDF:

1. It exists for all random variables (discrete, continuous, or mixed)
2. It completely determines the distribution
3. It can be estimated empirically from data
4. Probabilities are computed by subtraction rather than summation

Properties

Every CDF $F$ satisfies:

1. Non-decreasing

If $a < b$, then $F(a) \leq F(b)$.

Reason: As $x$ increases, we accumulate more probability.

2. Limits

$\lim_{x \to -\infty} F(x) = 0, \quad \lim_{x \to +\infty} F(x) = 1$

3. Right-Continuous

$\lim_{y \downarrow x} F(y) = F(x)$

For discrete RVs, this means the CDF jumps at the value, not before.

Bounds

$0 \leq F(x) \leq 1 \text{ for all } x$

Computing Probabilities

Using the CDF:

$P(a < X \leq b) = F(b) - F(a)$

$P(X > x) = 1 - F(x)$

$P(X < x) = F(x^-) \text{ (left-limit)}$

For discrete RVs with jump at $a$:

$P(X = a) = F(a) - F(a^-)$

The size of the jump at $a$ equals $P(X = a)$.

Example: Fair Die

For a fair die, the possible outcomes are $\\{1, 2, 3, 4, 5, 6\\}$.

$$F(x) = \begin{cases} 0 & x < 1 \\ \frac{1}{6} & 1 \leq x < 2 \\ \frac{2}{6} & 2 \leq x < 3 \\ \frac{3}{6} & 3 \leq x < 4 \\ \frac{4}{6} & 4 \leq x < 5 \\ \frac{5}{6} & 5 \leq x < 6 \\ 1 & x \geq 6 \end{cases}$$

This is a step function: flat between values, jumps at each possible outcome.

Example: Dice Sum

For sum of two dice, $X \in \{2, 3, \ldots, 12\}$.

$F(6) = P(X \leq 6) = \frac{1+2+3+4+5}{36} = \frac{15}{36} = \frac{5}{12}$

$F(7) = F(6) + P(X = 7) = \frac{15}{36} + \frac{6}{36} = \frac{21}{36} = \frac{7}{12}$

Relation to PMF

For discrete random variables:

$F(x) = \sum_{y \leq x} p(y)$

Conversely:

$p(x) = F(x) - F(x^-) = P(X = x)$

The PMF is the “derivative” of the CDF (in discrete sense).

Example: Geometric CDF

For $X \sim \text{Geo}(p)$ (geometric distribution):

$F(n) = P(X \leq n) = 1 - (1-p)^n$

Derivation: The event $X \leq n$ is equivalent to “at least one success in first $n$ trials”, which has probability $1 - P(\text{no successes}) = 1 - (1-p)^n$.

Percentiles

The $p$th percentile (or quantile) is the value $x_p$ such that:

$F(x_p) = p$

Or more carefully: $\inf\{x : F(x) \geq p\}$.

Example: The median is the 50th percentile: $F(x_{0.5}) = 0.5$.

Empirical CDF

Given data $x_1, x_2, \ldots, x_n$, the empirical CDF is:

$\hat{F}(x) = \frac{1}{n} \cdot |\{i : x_i \leq x\}|$

This is the proportion of observations at or below $x$.

By the Glivenko-Cantelli theorem, as $n \to \infty$, the empirical CDF converges uniformly to the true CDF.

Summary

Property	Description
Definition	$F(x) = P(X \leq x)$
Monotonicity	$a < b \Rightarrow F(a) \leq F(b)$
Limits	$F(-\infty) = 0$, $F(+\infty) = 1$
Jump size	$P(X = x) = F(x) - F(x^-)$
Interval prob	$P(a < X \leq b) = F(b) - F(a)$