CDF Calculator
Compute cumulative probabilities for Normal, Exponential, Uniform, Binomial, and Poisson distributions.
What is a cumulative distribution function (CDF)?
A cumulative distribution function tells you the probability that a random variable is less than or equal to a specific value. In notation, that is written as F(x) = P(X ≤ x). If you are studying statistics, risk, machine learning, quality control, or finance, the CDF is one of the most useful tools for understanding uncertainty.
This cumulative distribution function calculator gives you a fast way to compute those probabilities across common distributions without needing a statistics table.
How to use this calculator
- Select a distribution from the dropdown.
- Enter the threshold value (x, or k for discrete distributions).
- Provide the distribution parameters (for example μ and σ for Normal).
- Choose lower-tail probability P(X ≤ x) or upper-tail probability P(X > x).
- Click Calculate CDF.
Supported distributions and formulas
1) Normal distribution
For a normal random variable with mean μ and standard deviation σ:
Use this when your data is approximately bell-shaped (test scores, measurement error, many natural processes).
2) Exponential distribution
For rate λ > 0:
F(x) = 1 - e-λx, x ≥ 0
Useful for waiting-time problems, such as time until failure or time until the next event.
3) Uniform distribution (continuous)
For interval [a, b] with b > a:
F(x) = (x - a)/(b - a), a ≤ x ≤ b
F(x) = 1, x > b
Use this when every value in an interval is equally likely.
4) Binomial distribution
For number of trials n and success probability p:
Great for “how many successes out of n attempts?” questions.
5) Poisson distribution
For expected event rate λ:
Use this for counts of events in a fixed period or area (arrivals, defects, calls, etc.).
Example interpretations
- Normal: If F(1.5) = 0.9332, then about 93.32% of outcomes are ≤ 1.5.
- Binomial: If F(3) = 0.8150, then there is an 81.50% chance of at most 3 successes.
- Upper tail: If P(X > x) = 0.067, there is a 6.7% chance of exceeding x.
Why CDF is more useful than a single point probability
A probability at exactly one point may be tiny (or zero for continuous distributions). The CDF gives a cumulative view, which is what most real-world decisions need: pass/fail thresholds, risk limits, service-level guarantees, and anomaly detection cutoffs.
Common mistakes to avoid
- Using a negative standard deviation for Normal (σ must be positive).
- Using λ ≤ 0 for Exponential or Poisson (rate must be positive).
- Setting Uniform bounds incorrectly (must satisfy b > a).
- Entering non-integer trial counts for Binomial (n must be an integer).
- Mixing up lower tail and upper tail probabilities.
Quick FAQ
Is this calculator exact?
It uses standard numerical approximations and stable summation methods that are accurate for common practical use.
What is the difference between CDF and PDF?
The CDF gives cumulative probability up to x. The PDF/PMF gives density or mass at specific values.
Can I use this for hypothesis testing?
Yes. CDF and tail probabilities are core ingredients in p-value and critical-value calculations.