One-dimensional Discrete Distribution Functions

Class providing discrete probability distribution functions of a single integer variable—the probability mass function \(p(x)\) (and its logarithm), the cumulative distribution function \(P(x) = \sum_{x' \le x} p(x')\), and the quantile function \(x(P)\). These distributions model count data such as the number of galaxies in a halo or the number of star formation events, and are used for drawing random variates and computing Poisson or binomial likelihoods in galaxy statistics and N-body halo occupation analyses.

Default implementation: distributionFunctionDiscrete1DBinomial

Methods

mass → double precision

Return the probability mass function \(p(x)\), giving the probability that the discrete random variable takes the integer value x.

  • integer, intent(in ) :: x

massLogarithmic → double precision

Return the natural logarithm of the probability mass function \(\ln p(x)\) evaluated at integer x, which is more numerically stable for extremely small probabilities than computing \(p(x)\) directly.

  • integer, intent(in ) :: x

cumulative → double precision

Return the cumulative distribution function \(P(x) = \sum_{x' \le x} p(x')\), giving the probability that the discrete random variable takes a value less than or equal to integer x.

  • integer, intent(in ) :: x

  • integer, intent( out), optional :: status

cumulativeComplementary → double precision

Return the complementary cumulative probability at x.

  • integer, intent(in ) :: x

  • integer, intent( out), optional :: status

inverse → integer

Return the value of the independent variable corresponding to cumulative probability p.

  • double precision, intent(in ) :: p

sample → integer

Return a random integer deviate drawn from this discrete probability distribution, using the inverse CDF method by default (drawing a uniform random number and applying the quantile function).

  • class(randomNumberGeneratorClass), intent(inout), optional :: randomNumberGenerator_

minimum → integer

Returns the minimum possible integer value in the support of this discrete distribution, i.e., the smallest integer \(x\) for which the probability mass is non-zero.

maximum → integer

Returns the maximum possible integer value in the support of this discrete distribution, i.e., the largest integer \(x\) for which the probability mass is non-zero.

distributionFunctionDiscrete1DBinomial

A 1D binomial discrete distribution function class, modeling the number of successes \(k\) in \(n\) independent Bernoulli trials each with success probability \(p\), with probability mass function \(P(k) = \binom{n}{k} p^k (1-p)^{n-k}\).

(Default implementation)

Parameters

  • [probabilitySuccess] — The probability \(p \in [0,1]\) of success on a single Bernoulli trial, which determines the mean (\(np\)) and variance (\(np(1-p)\)) of the resulting binomial distribution.

  • [countTrials] — The total number of independent Bernoulli trials \(n\), which sets the range of the distribution from 0 to \(n\) and controls the overall scale of the mean and variance.

distributionFunctionDiscrete1DNegativeBinomial

A negative binomial 1D discrete distribution function class, modeling the number of successes \(k\) observed before a fixed number of failures \(r\) in a sequence of independent Bernoulli trials each with success probability \(p\), with probability mass function \(P(k) = \binom{k+r-1}{k} p^r (1-p)^k\). Note that the cumulative and inverse CDF methods are not currently implemented; only the probability mass function and its logarithm are available.

Parameters

  • [probabilitySuccess] — The probability \(p \in (0,1]\) of success on a single Bernoulli trial; the distribution models the number of successes before \(r\) failures occur, with mean \(pr/(1-p)\).

  • [countFailures] — The target number of failures \(r\) before the experiment stops; the distribution gives the number of successes \(k\) observed before the \(r\)-th failure, with variance \(pr/(1-p)^2\).

distributionFunctionDiscrete1DPoisson

A Poisson 1D discrete distribution function class.

Parameters

  • [mean] — The mean of the distribution.