Hypergeometric distribution Click on the name of a distribution to access the corresponding entry of the Glossary Click on a number to access the detailed Table of Contents of the Tutorial

 DISCRETE DISTRIBUTIONS   Binomial Geometric Hypergeometric Multinomial Negative binomial Poisson Poisson process

 Tutorials 1 Probability mass function. Mean, variance, mode.   Moment generating function. Generating function.    Additivity.    Convergence to a normal distribution.   Estimation of the two parameters n and p. 2 Demonstration of De Moivre theorem.
 Animation p and sample size adjustable.    Mean, mode, standard deviation.    Convergence to a normal distribution.

 Tutorial 1 Basic properties of the geometric distribution.    Geometric r.v. as discretized exponential r.v..
 Animation p and sample size adjustable.    Mean, mode, standard deviation.

 Tutorials 1 Basic properties of the hypergeometric distribution :        * Probability mass function (2 methods).       * Mean.        * Variance. 2 Two convergences of the hypergeometric distribution to a    binomial distribution :       * Fixed sample size.       * Fixed number of white balls. 3 A first sufficient condition for the hypergeometric distribution    to tend to a normal distribution.    A second, improved, sufficient condition.    A third, even better sufficient condition. 4 Distribution of two binomial rvs conditionally to their sum    is hypergeometric.
 Animations All parameters are adjustable.     Progressive build-up of the histogram. Distribution of two binomial rvs conditionally to their sum     is hypergeometric.     The two binomials are adjustable. Sum is adjustable.    Progressive build-up of the histogram.

 Tutorials 1 Basic properties of the multinomial distribution.    Multinomial coefficient (2 demonstrations).    Maximum Likelihood Estimation of the probabilities. 2 Covariance of two modalities:        * Direct calculation.        * By auxiliary Bernoulli variables.    The covariance matrix is not full rank.
 Tutorials 3 Merged categories.    Marginal distributions.    Conditional distributions. 4 Pearson's theorem:        * Basic goodness-of-fit test of the multinomial distribution.        * The asymptotic distribution of the Chi-square statistic           is χ2.

 Tutorial 1 Basic properties of the negative binomial distribution.
 Animation Probability p is adjustable. .  Progressive build-up of the histogram.

 Tutorial 1 Probability mass function (direct and by generating function).    Mean, variance (direct).    Moment generating function (direct and by limit of the binomial    moment generating function).    Normal limit for large values of the parameter. 2 Additivity property of Poisson random variables.    Splitting a Poisson variable.   Poisson variables conditionally to their sum are multinomial.
 Animations Parameter λ is adjustable. .  Comparison with binomial (n or p adjustable). Simulation by drawing observations from an exponential    distribution with adjustable parameter λ.

 Tutorials 1 Poisson distribution in an opening time interval.    Stationarity and independence of increments.    Second definition of a Poisson process. 2 Anti-bunching property.    Third definition of a Poisson process. 3 Covariance of two opening time intervals.    Event times of a PP. The order statistic property. 4 Superposition (or "merging") of Poisson processes.    Splitting (or "thinning") of a Poisson process.    Inhomogenous Bernoulli splitting.
 Animation Poisson process with adjustable rate λ.

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