Distributions.

Tune μ and σ; watch the bell shift and stretch against samples drawn live.

Count rare events with λ — and watch the bars converge to a Normal as λ grows.

n coins, probability p — see the discrete count distribution shift and stretch toward Gaussian.

Waiting times between Poisson events — and the only continuous memoryless distribution.

A distribution over probabilities. The Bayesian conjugate prior for coin flips.

An undefined mean, a meaningful median, and averages that retain the original heavy tails.

Count trials or failures until the first success; condition on the remaining wait.

A normal logarithm gives a positive, skewed variable. Compare original and log scales and compute upper tails.

Equal-length intervals have equal probability. Compute areas and invert the cumulative scale.

Waiting time for k Poisson events. Shape k controls skew; rate λ controls scale.

An event as a zero-or-one indicator: its probability, variance, and estimated proportion.

Flip until the r-th success. Dispersion that Poisson can’t match.

Enumerate finite samples without replacement and compute how dependence changes the count.

Heavier tails than Normal at small ν. Shrinks to a Gaussian as degrees of freedom grow.

Sum of k squared standard normals. The engine of variance tests.

Ratio of two chi-squareds. The ANOVA distribution.

Failure times. Shape k flips the hazard rate from falling to rising.

Power-law tails. The 80/20 rule lived on a log-log axis.

Two exponentials back-to-back. The prior behind L1 regularization.

Link continuous values, cumulative probabilities, odds, and quantiles.

The distribution of maxima. Extreme-value theory in one curve.

Compare bounded models and sensitivity to minimum, mode, and maximum.

Learn category probabilities on a simplex and integrate their uncertainty into future counts.

Connect covariance, conditional distributions, linear projections, and perfect-correlation limits.

Two bells, one weight. Unmixes into components you can steer by hand.

See how normalized sums retain their shape while ordinary averages shrink, hold, or widen.

Compare infinite rank probabilities with a finite cutoff, and inspect tail mass and moment thresholds.

Turn observations into exact cumulative probabilities, tied-value masses, and inverse quantiles.