Discrete Distributions
The functions described here are among the most commonly used discrete statistical distributions. You can compute their densities, means, variances, and other related properties. The distributions themselves are represented in the symbolic form
name[param_{1}, param_{2}, ...]. Functions such as
Mean, which give properties of statistical distributions, take the symbolic representation of the distribution as an argument.
"Continuous Distributions" describes many continuous statistical distributions.
BernoulliDistribution[p]  Bernoulli distribution with mean p 
BetaBinomialDistribution[,,n]  binomial distribution where the success probability is a BetaDistribution[, ] random variable 
BetaNegativeBinomialDistribution[,,n] 
 negative binomial distribution where the success probability is a BetaDistribution[, ] random variable 
BinomialDistribution[n,p]  binomial distribution for the number of successes that occur in n trials, where the probability of success in a trial is p 
DiscreteUniformDistribution[{i_{min},i_{max}}] 
 discrete uniform distribution over the integers from i_{min} to i_{max} 
GeometricDistribution[p]  geometric distribution for the number of trials before the first success, where the probability of success in a trial is p 
HypergeometricDistribution[n,n_{succ},n_{tot}] 
 hypergeometric distribution for the number of successes out of a sample of size n, from a population of size n_{tot} containing n_{succ} successes 
LogSeriesDistribution[]  logarithmic series distribution with parameter 
NegativeBinomialDistribution[n,p] 
 negative binomial distribution with parameters n and p 
PoissonDistribution[]  Poisson distribution with mean 
ZipfDistribution[]  Zipf distribution with parameter 
Discrete statistical distributions.
Most of the common discrete statistical distributions can be understood by considering a sequence of trials, each with two possible outcomes, for example, success and failure.
The Bernoulli distribution
BernoulliDistribution[p] is the probability distribution for a single trial in which success, corresponding to value 1, occurs with probability
p, and failure, corresponding to value 0, occurs with probability
1p.
The binomial distribution
BinomialDistribution[n, p] is the distribution of the number of successes that occur in
n independent trials, where the probability of success in each trial is
p.
The negative binomial distribution
NegativeBinomialDistribution[n, p] for positive integer
n is the distribution of the number of failures that occur in a sequence of trials before
n successes have occurred, where the probability of success in each trial is
p. The distribution is defined for any positive
n, though the interpretation of
n as the number of successes and
p as the success probability no longer holds if
n is not an integer.
The beta binomial distribution
BetaBinomialDistribution[, , n] is a mixture of binomial and beta distributions. A
BetaBinomialDistribution[, , n] random variable follows a
BinomialDistribution[n, p] distribution, where the success probability
p is itself a random variable following the beta distribution
BetaDistribution[, ]. The beta negative binomial distribution
BetaNegativeBinomialDistribution[, , n] is a similar mixture of the beta and negative binomial distributions.
The geometric distribution
GeometricDistribution[p] is the distribution of the total number of trials before the first success occurs, where the probability of success in each trial is
p.
The hypergeometric distribution
HypergeometricDistribution[n, n_{succ}, n_{tot}] is used in place of the binomial distribution for experiments in which the
n trials correspond to sampling without replacement from a population of size
n_{tot} with
n_{succ} potential successes.
The discrete uniform distribution
DiscreteUniformDistribution[{i_{min}, i_{max}}] represents an experiment with multiple equally probable outcomes represented by integers
i_{min} through
i_{max}.
The Poisson distribution
PoissonDistribution[] describes the number of events that occur in a given time period where
is the average number of events per period.
The terms in the series expansion of
log (1) about
=0 are proportional to the probabilities of a discrete random variable following the logarithmic series distribution
LogSeriesDistribution[]. The distribution of the number of items of a product purchased by a buyer in a specified interval is sometimes modeled by this distribution.
The Zipf distribution
ZipfDistribution[], sometimes referred to as the zeta distribution, was first used in linguistics and its use has been extended to model rare events.
PDF[dist,x]  probability mass function at x 
CDF[dist,x]  cumulative distribution function at x 
InverseCDF[dist,q]  the largest integer x such that CDF[dist, x] is at most q 
Quantile[dist,q]  q^{th} quantile 
Mean[dist]  mean 
Variance[dist]  variance 
StandardDeviation[dist]  standard deviation 
Skewness[dist]  coefficient of skewness 
Kurtosis[dist]  coefficient of kurtosis 
CharacteristicFunction[dist,t]  characteristic function (t) 
ExpectedValue[f,dist]  expected value of the pure function f in dist 
ExpectedValue[f[x],dist,x]  expected value of f[x] for x in dist 
RandomInteger[dist]  pseudorandom number with specified distribution 
RandomInteger[dist,dims]  pseudorandom array with dimensionality dims, and elements from the specified distribution 
Functions of statistical distributions.
Distributions are represented in symbolic form.
PDF[dist, x] evaluates the mass function at
x if
x is a numerical value, and otherwise leaves the function in symbolic form whenever possible. Similarly,
CDF[dist, x] gives the cumulative distribution and
Mean[dist] gives the mean of the specified distribution. For a more complete description of the various functions of a statistical distribution, see the description of their continuous analogues in
"Continuous Distributions".
Here is a symbolic representation of the binomial distribution for 34 trials, each having probability 0.3 of success.
Out[1]=  

This is the mean of the distribution.
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You can get the expression for the mean by using symbolic variables as arguments.
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Here is the 50% quantile, which is equal to the median.
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This gives the expected value of x^{3} with respect to the binomial distribution.
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The elements of this matrix are pseudorandom numbers from the binomial distribution.
Out[6]=  
