# Cumulative distribution function

The cumulative distribution function (CDF) gives P(X <= x), the chance the variable is at most x. You get it by adding the probabilities up to x. It never decreases and runs from 0 to 1.

Statistics (university) · Part 03: Random variables and discrete distributions · Card 8 of 8 · Last updated 2026-10-19

## Analogy

A running total: after each step you record how much probability you have collected so far.

## Example

**Three coin flips**

P(0)=0.125, P(1)=0.375
P(2)=0.375, P(3)=0.125
P(X <= 1) = 0.125 + 0.375 = 0.5
P(X >= 2) = 1 - 0.5 = 0.5

## In practice

Tables for binomial and Poisson usually show the CDF, so you can answer 'at most' and 'at least' questions.

## Practice

### Flashcards

- **What does the CDF give?** P(X <= x), the chance of at most x.

## Questions people ask

### How do I find 'at least' probabilities from a CDF?

Use the complement. P(X at least k) equals 1 minus P(X at most k-1). For three flips, P(at least 2 heads) is 1 minus P(at most 1), which is 1 minus 0.5, so 0.5. This is faster than adding many terms.

### Is P(X < 3) the same as P(X <= 3) for discrete variables?

No. For discrete variables the single value matters, so P(X <= 3) includes P(X = 3) while P(X < 3) does not. For integer values P(X < 3) equals P(X <= 2). For continuous variables the two are equal because single points carry no probability.

## Related concepts

- [Binomial distribution](https://studyvisually.com/learn/science/statistics-university/statistics-university-part-3/statistics-university-part-3-concept-5)
- [Poisson distribution](https://studyvisually.com/learn/science/statistics-university/statistics-university-part-3/statistics-university-part-3-concept-6)
- [Geometric distribution](https://studyvisually.com/learn/science/statistics-university/statistics-university-part-3/statistics-university-part-3-concept-7)

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