Probability Distribution Calculator - Normal, Binomial & Poisson

How to use this tool

Select a distribution type: Normal, Binomial, or Poisson.
Enter the distribution parameters (mean, σ, n, p, or λ); binomial p and Poisson λ may be zero.
Enter a query value x (or a whole-number k for discrete distributions) to get PDF/PMF and CDF probabilities.

Common probability distributions

Distribution	Use case	Key parameters
Normal (Gaussian)	Heights, test scores, measurement errors	Mean (μ), standard deviation (σ)
Binomial	Coin flips, pass/fail outcomes in n trials	n (trials), p (probability of success)
Poisson	Rare events in fixed time/space (calls per hour)	λ (average rate)
Uniform	Equal probability outcomes (dice roll)	a (min), b (max)
Exponential	Time between events (wait time)	λ (rate)

The 68-95-99.7 rule

For a normal distribution: 68% of values fall within ±1σ of the mean, 95% within ±2σ, and 99.7% within ±3σ. This is also called the empirical rule.

How discrete endpoint values are handled

Binomial and Poisson distributions count whole-number outcomes, so this tool requires whole numbers for n and k. Their boundary values are valid: with a binomial probability of 0, exactly zero successes has probability 1; with probability 1, exactly n successes has probability 1. Likewise, a Poisson process with rate λ = 0 has probability 1 of zero events. These cases are evaluated directly instead of relying on logarithms such as log(0), which are undefined.

Choosing and interpreting a distribution

Use the binomial model only when there is a fixed number of independent trials with the same success probability. Use Poisson for a count over a fixed interval when its rate is reasonably constant. The normal curve is continuous, so its PDF is a density rather than the probability of one exact value; probabilities over ranges come from differences between CDF values. The chart and all calculations run locally in your browser.

Frequently asked questions

What is the Normal distribution?

A continuous bell-shaped distribution defined by mean (μ) and standard deviation (σ). The CDF gives the probability that a random variable is ≤ x.

When to use Poisson vs Binomial?

Use Binomial for a fixed number of independent trials with success probability p. Use Poisson when counting rare events in a fixed time/space interval.

Can p or λ be zero?

Yes. For a binomial distribution, p = 0 means zero successes is certain and p = 1 means n successes is certain. For a Poisson distribution, λ = 0 means zero events is certain. These valid degenerate cases are calculated directly.