Probability Distributions

Choose a probability distribution (normal, binomial, Poisson, exponential, uniform), set its parameters, and draw random samples. Watch the histogram fill in and settle onto the theoretical shape as the sample grows.

Speed

How the model works

Normal (μ, σ): the bell curve. About 68% of values fall within 1σ and 95% within 2σ.
Binomial (n, p): the number of successes in n independent trials. Mean np, variance np(1 − p).
Poisson (λ): counts of rare, independent events in a fixed interval. Mean and variance both λ.
Exponential (λ): waiting time between Poisson events. Mean 1/λ; memoryless.
Uniform (a, b): every value in [a, b] equally likely. Mean (a + b)/2, variance (b − a)²/12.

The histogram is scaled to a density (or probability for discrete distributions), so it can be compared directly with the theoretical curve. Sample statistics wander for small samples and converge as the sample grows (the law of large numbers). Samples use a seeded generator, so a run can be reproduced exactly.

Normal samples use the Box–Muller method; binomial and Poisson use direct simulation for modest parameters.

Guided experiments

68–95–99.7. Standard normal, 5000 samples. Prediction: about 68% within ±1, 95% within ±2. See the readout.
Binomial looks normal. n = 100, p = 0.5. Prediction: an almost perfect bell centred on 50 with σ = 5.
Skewed Poisson. λ = 2. Prediction: right-skewed; P(0) = e⁻² ≈ 13.5%. The mean and variance both come out near 2.

Challenge

A call centre gets on average 3 calls per minute. What is the probability of no calls in a given minute, and what is the average gap between calls?

Reveal the explanation

Calls per minute are Poisson(3): P(0) = e⁻³ ≈ 4.98%. The gaps are exponential with λ = 3 per minute, so the mean gap is 1/3 minute, 20 seconds. Check both with the Poisson and exponential settings.

References

OpenStax, Introductory Statistics, ch. 4–6.