Day 1 · Interactive

Discrete distribution explorer

Drag the sliders and watch the probability mass function change. The goal is to feel how each distribution's story (its parameters) controls its shape, mean, and variance.

Show the plotted values as a table

Tail probabilities

Draw a sample and compare it with the theory

No sample drawn yet. The hollow bars will show the sample proportions on top of the theoretical pmf.

Things to try

  1. Binomial → symmetric vs skewed. Set p = 0.5 and watch the PMF become symmetric; push p toward 0 or 1 and it skews.
  2. Poisson mean = variance. Change λ and note the mean and variance readouts stay equal. This is the Poisson's defining constraint, and the reason it so often fails on real ecological counts.
  3. Overdispersion. Switch to Negative Binomial, tick "overlay a Poisson with the same mean," and drag the dispersion k down. The NegBin spreads far wider than the Poisson at the same mean. That extra spread is what clumped ecological data need.
  4. NegBin → Poisson limit. Drag k up to its maximum; the Negative Binomial collapses onto the Poisson.
  5. Both tails at once. On the Negative Binomial with μ = 10, set the threshold to c = 20 and drag k down. P(X = 0) and P(X > 20) rise together. That is the "many zeros and a few huge counts" signature of clumped data.
  6. How noisy is a sample variance? Draw n = 100 values a few times and watch the sample variance jump around while the sample mean barely moves, especially at small k. Then raise n to 2000. This is the "mind the gap" lesson from the whale exercise in the lab.

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