Distribution Function — Definition, Formula & Examples

A distribution function (also called the cumulative distribution function or CDF) is a function that gives the probability that a random variable takes a value less than or equal to a given number. For any real number

x

, it tells you

F(x) = P(X \le x)

The (cumulative) distribution function of a random variable $X$ is the function $F : \mathbb{R} \to [0,1]$ defined by $F(x) = P(X \le x)$ for all $x \in \mathbb{R}$ . It is non-decreasing, right-continuous, satisfies $\lim_{x \to -\infty} F(x) = 0$ , and $\lim_{x \to +\infty} F(x) = 1$ .

Key Formula

F(x) = P(X \le x)

Where:

$F(x)$ = The probability that the random variable X takes a value at most x
$X$ = A random variable
$x$ = Any real number at which the CDF is evaluated

How It Works

To find

F(x)

for a discrete random variable, sum the probabilities of all outcomes less than or equal to

x

. For a continuous random variable, integrate the probability density function from

-\infty

x

. The value

F(x)

always lies between 0 and 1, and it never decreases as

x

increases. You can also find the probability that

X

falls in an interval

(a, b]

by computing

F(b) - F(a)

Worked Example

Problem: A fair die is rolled once. Let X be the number shown. Find the distribution function F(x) and compute P(X ≤ 4).

Step 1: List the probability of each outcome. Each face has probability 1/6.

P(X = k) = \frac{1}{6} \quad \text{for } k = 1, 2, 3, 4, 5, 6

Step 2: The CDF at x = 4 is the sum of probabilities for all outcomes ≤ 4.

F(4) = P(X \le 4) = \frac{1}{6} + \frac{1}{6} + \frac{1}{6} + \frac{1}{6} = \frac{4}{6}

Step 3: Simplify the fraction.

F(4) = \frac{2}{3} \approx 0.667

Answer: F(4) = 2/3, so there is approximately a 66.7% probability that the die shows 4 or less.

Why It Matters

Distribution functions are central to hypothesis testing, confidence intervals, and reliability engineering. When you look up a p-value in a z-table or t-table, you are reading values from a CDF. Any probability question about a random variable can ultimately be answered using its distribution function.

Common Mistakes

Mistake: Confusing the distribution function (CDF) with the probability mass function (PMF) or probability density function (PDF).

Correction: The CDF gives cumulative probability P(X ≤ x), which is a running total. The PMF gives P(X = x) for discrete variables, and the PDF gives the density (not probability) at a point for continuous variables. The CDF is obtained by summing the PMF or integrating the PDF.