## Statistical Distributions

**numerics4c++** provides the means to compute the cumulative
distribution function (CDF) probabilities for common distributions. Along with CDF computations, **numerics4c++**
provides for the evaluation of inverse CDF values. Additionally, **numerics4c++**
provided the means to compute the probability mass function (PMF) probabilities for discrete deistributions.

To use a distribution, simply create a new distribution object with the desired parameters. With the distribution object, call the cdf function to compute CDF probabilities, the inverse_cdf function to compute inverse CDF values and, the pmf function to compute PMF probabilities:

// create a distribution with the desired parameters normal_distribution z(0.0, 1.0); double p = z.cdf(1.96); // p = P(z <= 1.96) double x = z.inverse_cdf(0.975); // P(z <= x) = 0.975 // create a discrete distribution binomial_distribution b(10, 0.5); p = b.pmf(4); // p = P(b = 4)

The following distribution are supplied by **numerics4c++**:

Distribution | Class | Parameters |
---|---|---|

Beta | beta_distribution | alpha beta |

Binomial | binomial_distribution | number of trials probability of success |

Cauchy | cauchy_distribution | location scale |

Chi-Squared | chi_squared_distribution | degrees of freedom |

Exponential | exponential_distribution | mean |

F | f_distribution | numerator degrees of freedom denominator degrees of freedom |

Gamma | gamma_distribution | alpha beta |

Geometric | geometric_distribution | probability of success |

Hypergeometric | hypergeometric_distribution | population number of successes population number of failures sample size |

Laplace | laplace_distribution | mean scale |

Logistic | logistic_distribution | mean scale |

Log Normal | log_normal_distribution | mean standard deviation |

Negative Binomial | negative_binomial_distribution | number of successes probability of success |

Normal | normal_distribution | mean standard deviation |

Poisson | poisson_distribution | mean |

Rayleigh | rayleigh_distribution | scale |

t | t_distribution | degrees of freedom |

Uniform | uniform_distribution | lower bound upper bound |

### Inverse CDF's for Discrete Distributions

Special care is given to computing the inverse CDF for discrete distributions. This is because of the inexactness of the mapping from cumulative probabilities to quantiles. As such, the return value, x, of the inverse_cdf method satifies the following to criteria:

For the distribution X and target probability p: x is the largest integer such that P(X <= x) <= p and x is the smallest integer such that P(X >= x) >= 1 - pFor example, for a Binomial(10, 0.375) distribution:

num::binomial_distribution b(10, 0.375); double p = 0.75; double x = b.inverse_cdf(p); // assert: b.cdf(x) <= 0.75 // assert: b.cdf(x + 1) >= 0.75 // assert: 1.0 - b.cdf(x) >= 0.25 // assert: 1.0 - b.cdf(x + 1) <= 0.25