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A probability distribution describes the relative likelihood of obtaining each possible outcome from a random process. For example, the distribution of heights in a group of people (measured as deviations from the mean), might be described
by the distribution pictured below.

Meta.Numerics provides you with a large number of commonly used probability distributions. Each probability distribution is represented by a class in the Meta.Numerics.Statistics.Distributions namespace. Continuous distributions, such as NormalDistribution, inherit from

the Distribution class. Discrete distributions, such as BinomialDistribution, inherit from the DiscreteDistribution class.

### Summary Data

For any distribution, you can obtain many useful summary statistics.

### Random Deviates

Given any distribution, you can use the .NET Framework's Random class to obtain random number drawn from that distribution.

### Provided Distributions

The following continuous probability distributions are provided by the Meta.Numerics library:

All the continuous distribution classes inherit from the abstract Distribution class.

### Example

Given a distribution, the following code computes many of its properties by numerical integration and compares them with the values returned by the member functions.

Meta.Numerics provides you with a large number of commonly used probability distributions. Each probability distribution is represented by a class in the Meta.Numerics.Statistics.Distributions namespace. Continuous distributions, such as NormalDistribution, inherit from

the Distribution class. Discrete distributions, such as BinomialDistribution, inherit from the DiscreteDistribution class.

Member | Description |
---|---|

Mean | The mean of the distribution. |

Median | The median of the distribution. |

StandardDeviation | The standard deviation of the distribution. |

Variance | The variance of the distribution. This is the square of the standard deviation. |

Skewness | The skewness of the distribution. Symmetric distributions have zero skew. |

Moment(n) | The nth raw moment of the distribution. |

MomentAboutMean(n) | The nth central moment of the distribution. |

Support | The interval over which the distribution is non-vanishing. |

ProbabilityDensity(x) | The PDF of the distribution at the given point. |

LeftProbability(x) | The CDF of the distribution at the given point, i.e. the probability that a value is less than x. This is the area under the PDF to the left of x. |

RightProbability(x) | The complementary CDF of the distribution at the given point, i.e. the probability that a value is greater than x. This is the area under the PDF to the right of x. |

InverseLeftProbability(P) | The inverse of the CDF. Given a percentile rank P, this function returns the score x for which a fraction P of values are lower. |

// Write out 20 exponentially distributed values Random rng = new Random(); Distribution d = new ExponentialDistribution(); for (int i = 0; i < 20; i++) { Console.WriteLine(d.GetRandomValue(rng)); }

Distribution Class | Description |
---|---|

NormalDistribution | The normal, or Gaussian distribution, appears in the law-of-large-numbers as an approximation to many probabilistic properties. It is commonly used as an approximate model for any symmetric, unimodular distribution. |

LognormalDistribution | The distribution of values whose logrithms are normally distributed. It is commonly used in financial mathematics to represent the distribution of values of a security whose percent change is subject to Gaussian fluctuations. |

ExponentialDistribution | The exponential distribution is the result of any constant-hazard process, such as particle decay. |

WeibullDistribution | The Weibull distribution is a generalization of the exponenial distribution, which allows for a hazard that increases or decreases over time. It is often used as a model for time-to-failure data. |

UniformDistribution | The uniform distribution represents a process that is equally likely to produce a value anywhere in a given interval. |

TriangularDistribution | A distribution on an arbitrary interval with a triangular PDF. |

BetaDistribution | A distribution on the unit interval with a flexible shape. |

ChiSquaredDistribution | The distribution of a sum of squares of n normal deviates. This is the distribution of the chi squared goodness-of-fit statistic. |

FisherDistribution | The distribution of a ratio of two chi-squared deviates. This is the distribution of the Fisher F statistic. |

StudentDistribution | The distribution of the Student t statistic. |

KolmogorovDistribution | The distribution of the Kolmogorov-Smirnov D statistic. |

All the continuous distribution classes inherit from the abstract Distribution class.

public void TestMoments (Distribution d) { // the support gives the limits of integration Interval support = d.Support; // raw moments double[] M = new double[6]; for (int n = 0; n < 6; n++) { // define x^n p(x) Function<double, double> raw = delegate(double x) { return (Math.Pow(x, n) * d.ProbabilityDensity(x)); }; // integrate it M[n] = FunctionMath.Integrate(raw, support); // compare with the claimed result Console.WriteLine("M{0} {1} v. {2}", n, M[n], d.Moment(n)); } // central moments double[] C = new double[6]; for (int n = 0; n < 6; n++) { // define (x-m)^n p(x) Function<double, double> central = delegate(double x) { return (Math.Pow(x - M[1], n) * d.ProbabilityDensity(x)); }; // integrate it C[n] = FunctionMath.Integrate(central, support); // compare with the claimed result Console.WriteLine("C{0} {1} v. {2}", n, C[n], d.MomentAboutMean(n)); } Console.WriteLine("Mean {0} v. {1}", M[1], d.Mean); Console.WriteLine("Standard Deviation {0} v. {1}", Math.Sqrt(C[2]), d.StandardDeviation); }

Last edited Apr 2, 2011 at 12:52 AM by ichbin, version 6