What Is Cumulative Distribution?
The cumulative distribution function, or CDF, is the probability that a random variable is less than or equal to x. It adds up all the probability from the far left up to x, so it rises from 0 to 1.
Choose a distribution, enter its parameters and a value x. You get the CDF, the upper tail and the density or probability at x, along with a graph of the CDF.
To find the probability between two values, calculate the CDF at each and subtract.
How to Use the Cumulative Distribution Calculator
- Choose the distribution.
- Enter its parameters and the value x.
- Read P(X ≤ x), the upper tail and the density or probability at x.
Cumulative Distribution Formula
CDF
Interval
Cumulative Distribution Variables Explained
| Symbol | Meaning | Notes |
|---|---|---|
| F(x) | Cumulative distribution function | P(X ≤ x) |
| f(x) | Density, or the probability at x for discrete variables | |
| 1 − F(x) | Upper tail probability | |
| F(b) − F(a) | Probability between a and b | For a < b |
Cumulative Distribution Worked Example
With the example values loaded in the calculator above, the steps are:
- Distribution: Binomial(n = 10, p = 0.5).
- The CDF is F(x) = P(X ≤ x). Evaluate it at x = 6: F(6) = 0.828125.
- P(X ≥ x) = 1 − P(X ≤ 5) = 0.376953.
Result: P(X ≤ x), the CDF = 0.828125; P(X ≥ x) = 0.376953.
Worked example: Binomial(10, 0.5), x = 6
Add the probabilities of 0 through 6 successes. P(X ≤ 6) = 0.828125.
The upper tail P(X ≥ 6) = 1 − P(X ≤ 5) = 1 − 0.623047 = 0.376953.
Discrete and continuous
For a discrete variable such as a binomial count the CDF steps up at each whole number, and P(X ≥ x) includes x. For a continuous variable P(X = x) is zero, so P(X ≥ x) and P(X > x) are equal.
Where Cumulative Distribution Is Used
Probability homework
Find P(X ≤ x) and P(X ≥ x) for common distributions.
Binomial and Poisson counts
Add up the probabilities of several counts at once.
Reading the CDF graph
See how probability accumulates as x increases.
Common Cumulative Distribution Mistakes to Avoid
- For a range, subtract two CDF values.
- The density is not a probability for a continuous variable.
Cumulative Distribution Terms and Definitions
- CDF
- The function F(x) = P(X ≤ x).
- PMF
- The probability mass function, P(X = x), for a discrete variable.
Cumulative Distribution FAQs
What is a cumulative distribution function?
A function that gives P(X ≤ x). It starts at 0, ends at 1 and never decreases.
How do I find P(X > x)?
Subtract the CDF from 1. For a discrete variable, P(X ≥ x) is 1 minus the CDF at the value just below x.
How do I find the probability between two values?
Subtract the CDF at the lower value from the CDF at the upper value.
What is the difference between PDF and CDF?
The PDF or PMF gives density or probability at a point. The CDF gives the accumulated probability up to that point.