CDF: Cumulative Distribution Function, Formula, Examples, and Interpretation

Think of the CDF as asking: "How much probability has accumulated up to this point?" It gives you a running total as x moves from left to right across all possible values.

What Is a Cumulative Distribution Function?

Suppose X represents the number of customers entering a shop in one hour. If F(20) = 0.80, that means P(X ≤ 20) = 0.80 — there's an 80% chance that 20 or fewer customers arrive. The CDF doesn't tell you the probability of exactly 20 customers; it tells you the probability of at most 20.

This accumulation property is what makes the CDF so useful. Rather than looking at probability piece by piece, the CDF gives you the full picture up to any chosen value.

The CDF applies to every type of random variable — discrete (like counts) and continuous (like heights or weights). This makes it one of the most general tools in probability and statistics.

CDF Formula

The formula depends on whether X is discrete or continuous.

CDF for Discrete Random Variables

For a discrete random variable, the CDF is built by adding up probabilities one value at a time. At each possible value, you accumulate what has come before.

The discrete CDF has a step-function shape: it stays flat between possible values and jumps at each one. The size of each jump equals the probability of that exact value.

Worked Example — Discrete CDF

CDF for Continuous Random Variables

For a continuous random variable, the CDF is obtained by integrating the probability density function (PDF) from negative infinity up to x. The key fact about continuous random variables: the probability of any single exact value is zero. P(X = x) = 0 for every x. Only intervals carry probability.

Worked Example — Continuous CDF (Uniform Distribution)

Key Properties of a CDF

Every valid cumulative distribution function satisfies five mathematical properties. These aren't optional — they define what a CDF is.

How to Calculate Probabilities Using a CDF

The CDF is the workhorse of probability calculations. Here are the four standard formulas you'll use constantly.

Strict vs Non-Strict Inequalities

This distinction trips up many students. For a continuous random variable, P(X = x) = 0 for every single point, so the inequality sign doesn't matter:

For a discrete random variable, they can differ. If P(X = 3) = 0.15, then P(X ≤ 3) = F(3) but P(X < 3) = F(2), which is smaller by 0.15.

Interactive CDF Visualizer

Select a distribution and drag the x-value slider to see the CDF in action. The shaded region shows P(X ≤ x).

CDF vs PDF

The CDF and PDF both describe a probability distribution, but from different angles. The CDF gives cumulative probability; the PDF gives probability density.

The key mathematical relationship between the two:

CDF vs PMF

PMF vs PDF vs CDF — Full Comparison

CDF of Common Probability Distributions

Normal Distribution

Uniform Distribution

Exponential Distribution

Binomial Distribution

Poisson Distribution

Empirical CDF (ECDF)

When you have real data, you can estimate the population CDF directly from your observations. This is the empirical CDF, or ECDF.

For a sample of n observations x₁, x₂, ..., xₙ, the ECDF at any point x is the fraction of observations that fall at or below x:

The ECDF is a step function — it jumps at every data point. It's completely nonparametric: no distribution assumption needed.

CDF, Quantiles, and Percentiles

The CDF and quantiles are two sides of the same coin. The CDF maps a value to a probability; a quantile maps a probability back to a value.

To find the 90th percentile: solve F(x) = 0.90 for x. For the median: solve F(m) = 0.50. This is exactly what software functions like qnorm() in R or norm.ppf() in Python compute. Learn more about z-scores and percentiles.

Survival Function

The survival function S(x) is the complement of the CDF:

If X represents the lifetime of a machine component, S(t) gives the probability that the component is still working at time t. Survival functions appear throughout reliability engineering, insurance, and clinical trials.

CDF Calculator

CDF in Real-World Applications

CDF in Statistical Software

CDF in R

In R, the p prefix gives the CDF for any distribution family.

# Normal CDF: P(X ≤ 1.5) where X ~ N(0,1)
pnorm(1.5, mean = 0, sd = 1)        # → 0.9332

# Uniform CDF: P(X ≤ 3) where X ~ Uniform(0,10)
punif(3, min = 0, max = 10)          # → 0.3000

# Exponential CDF: P(X ≤ 2) where X ~ Exp(rate=1)
pexp(2, rate = 1)                   # → 0.8647

# Binomial CDF: P(X ≤ 4) where X ~ Binom(10, 0.5)
pbinom(4, size = 10, prob = 0.5)    # → 0.3770

# Poisson CDF: P(X ≤ 3) where X ~ Poisson(lambda=2)
ppois(3, lambda = 2)                 # → 0.8571

CDF in Python

from scipy import stats

# Normal CDF: P(X ≤ 1.5) where X ~ N(0,1)
stats.norm.cdf(1.5, loc=0, scale=1)       # → 0.9332

# Exponential CDF: P(X ≤ 2) where X ~ Exp(rate=1)
stats.expon.cdf(2, scale=1)               # → 0.8647

# Binomial CDF: P(X ≤ 4) where X ~ Binom(10, 0.5)
stats.binom.cdf(4, n=10, p=0.5)           # → 0.3770

# Poisson CDF: P(X ≤ 3) where X ~ Poisson(lambda=2)
stats.poisson.cdf(3, mu=2)                # → 0.8571

CDF in Excel

Common Mistakes with CDFs

Frequently Asked Questions

Related Statistics Resources