🎲 احتمال و آمار

Monte Carlo πEstimate π by throwing random darts at a unit square with an inscribed circle. The ratio… Central Limit TheoremDraw repeated samples from Uniform, Exponential, Bimodal or Poisson distributions and watch… Markov ChainsEdit a transition matrix, step a random walker between states, and watch the distribution… Markov Chain — Transition Matrix SimulatorEdit the transition matrix and watch state probabilities converge to a stationary… Percolation Theory — Site & Bond Percolation, Phase TransitionInteractive percolation simulation on a 2D lattice. Adjust the occupation probability p and… Law of Large NumbersNewRoll dice thousands of times and watch the running average converge to the expected value… Gambler's RuinSet stake, target and win probability, then run Monte Carlo random walks to see how often… Stochastic ProcessesSample Brownian motion, geometric Brownian motion and Ornstein-Uhlenbeck paths, and see how… Lévy FlightSimulate Lévy flights: tune the stability exponent α from Brownian motion (α=2) to rare long… Stochastic ProcessesNewRandom walks, Brownian motion, geometric Brownian motion, Poisson processes and the central… Buffon's Needle — Monte Carlo π EstimationNewEstimate π by dropping needles on a lined floor: π ≈ 2L·N / (d·crossings). Watch the… Coupon Collector's Problem — Expected Draws to Collect AllNewCollect all N coupons from random draws: watch the grid fill, compare empirical averages to… Secretary Problem — Optimal Stopping & the 37% RuleNewReject the first ~37% of candidates, then hire the first one better than all seen.… Hidden Markov Model — States & ObservationsNewWatch a hidden Markov model emit observations from latent states, then recover the most… Reservoir Sampling — Fair Samples from a StreamNewPick k uniform samples from a stream of unknown length in one pass: item i replaces a… Dynamic Time Warping — Aligning Time SeriesNewDynamic time warping aligns two time series that vary in speed by warping the time axis. A… Benford's LawCount leading digits in Fibonacci numbers, populations or your own data and see how closely… Regression to the MeanPick extreme performers from a correlated pair of scores and watch their follow-up drift… Monte Carlo Importance SamplingNewEstimate rare-event probabilities with importance sampling: shift the proposal distribution…