The functions described here are among the most commonly used discrete univariate statistical distributions. You can compute their densities, means, variances, and other related properties. The distributions themselves are represented in the symbolic form
. 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|
|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|
|discrete uniform distribution over the integers from to |
|GeometricDistribution[p]||geometric distribution for the number of trials before the first success, where the probability of success in a trial is p|
|hypergeometric distribution for the number of successes out of a sample of size n, from a population of size containing 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
, and failure, corresponding to value 0, occurs with probability
The binomial distribution BinomialDistribution
is the distribution of the number of successes that occur in
independent trials, where the probability of success in each trial is
The negative binomial distribution NegativeBinomialDistribution
for positive integer
is the distribution of the number of failures that occur in a sequence of trials before
successes have occurred, where the probability of success in each trial is
. The distribution is defined for any positive
, though the interpretation of
as the number of successes and
as the success probability no longer holds if
is not an integer.
The beta binomial distribution BetaBinomialDistribution
is a mixture of binomial and beta distributions. A BetaBinomialDistribution
random variable follows a BinomialDistribution
distribution, where the success probability
is itself a random variable following the beta distribution BetaDistribution
. The beta negative binomial distribution BetaNegativeBinomialDistribution
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
The hypergeometric distribution HypergeometricDistribution
is used in place of the binomial distribution for experiments in which the
trials correspond to sampling without replacement from a population of size
The discrete uniform distribution DiscreteUniformDistribution
represents an experiment with multiple equally probable outcomes represented by integers
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
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 is at most q|
|Quantile[dist,q]||q quantile |
|StandardDeviation[dist]||standard deviation |
|Skewness[dist]||coefficient of skewness |
|Kurtosis[dist]||coefficient of kurtosis |
|CharacteristicFunction[dist,t]||characteristic function |
|Expectation[f[x],xdist]||expectation of for x distributed according to dist|
|Quartiles[dist]||list of the , , quantiles for dist|
|InterquartileRange[dist]||difference between the first and third quartiles|
|QuartileDeviation[dist]||half the interquartile range|
|QuartileSkewness[dist]||quartile-based skewness measure|
|RandomVariate[dist]||pseudorandom number with specified distribution |
|RandomVariate[dist,dims]||pseudorandom array with dimensionality dims, and elements from the specified distribution |
Some functions of statistical distributions.
Distributions are represented in symbolic form. PDF
evaluates the mass function at x
is a numerical value, and otherwise leaves the function in symbolic form whenever possible. Similarly, CDF
gives the cumulative distribution and Mean[dist]
gives the mean of the specified distribution. The table above gives a sampling of some of the more common functions available for distributions. For a more complete description of these functions, 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.
This is the mean of the distribution.
You can get the expression for the mean by using symbolic variables as arguments.
Here is the 50% quantile, which is equal to the median.
This gives the expected value of
with respect to the binomial distribution.
The elements of this matrix are pseudorandom numbers from the binomial distribution.