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TDistribution Class |
Inheritance HierarchyAccord.Statistics.DistributionsDistributionBase
Accord.Statistics.Distributions.UnivariateUnivariateContinuousDistribution
Accord.Statistics.Distributions.UnivariateTDistribution
Namespace: Accord.Statistics.Distributions.Univariate
Assembly: Accord.Statistics (in Accord.Statistics.dll) Version: 3.8.0
Syntax[SerializableAttribute] public class TDistribution : UnivariateContinuousDistribution, IFormattable
The TDistribution type exposes the following members.
Constructors| Name | Description | |
|---|---|---|
 | TDistribution |
Initializes a new instance of the TDistribution class.
|
Properties| Name | Description | |
|---|---|---|
 | DegreesOfFreedom |
Gets the degrees of freedom for the distribution.
|
| Entropy |
Not supported.
(Overrides UnivariateContinuousDistributionEntropy.) |
| Mean |
Gets the mean for this distribution.
(Overrides UnivariateContinuousDistributionMean.) |
| Median |
Gets the median for this distribution.
(Inherited from UnivariateContinuousDistribution.) |
| Mode |
Gets the mode for this distribution (always zero).
(Overrides UnivariateContinuousDistributionMode.) |
| Quartiles |
Gets the Quartiles for this distribution.
(Inherited from UnivariateContinuousDistribution.) |
| StandardDeviation |
Gets the Standard Deviation (the square root of
the variance) for the current distribution.
(Inherited from UnivariateContinuousDistribution.) |
| Support |
Gets the support interval for this distribution.
(Overrides UnivariateContinuousDistributionSupport.) |
| Variance |
Gets the variance for this distribution.
(Overrides UnivariateContinuousDistributionVariance.) |
Methods| Name | Description | |
|---|---|---|
| Clone |
Creates a new object that is a copy of the current instance.
(Overrides DistributionBaseClone.) |
| ComplementaryDistributionFunction |
Gets the complementary cumulative distribution function
(ccdf) for this distribution evaluated at point x.
This function is also known as the Survival function.
(Inherited from UnivariateContinuousDistribution.) |
| CumulativeHazardFunction |
Gets the cumulative hazard function for this
distribution evaluated at point x.
(Inherited from UnivariateContinuousDistribution.) |
| DistributionFunction(Double) |
Gets the cumulative distribution function (cdf) for
this distribution evaluated at point x.
(Inherited from UnivariateContinuousDistribution.) |
| DistributionFunction(Double, Double) |
Gets the cumulative distribution function (cdf) for this
distribution in the semi-closed interval (a; b] given as
P(a < X ≤ b).
(Inherited from UnivariateContinuousDistribution.) |
| Equals | Determines whether the specified object is equal to the current object. (Inherited from Object.) |
 | Finalize | Allows an object to try to free resources and perform other cleanup operations before it is reclaimed by garbage collection. (Inherited from Object.) |
 | Fit(Double) |
Fits the underlying distribution to a given set of observations.
(Inherited from UnivariateContinuousDistribution.) |
| Fit(Double, IFittingOptions) |
Fits the underlying distribution to a given set of observations.
(Inherited from UnivariateContinuousDistribution.) |
| Fit(Double, Double) |
Fits the underlying distribution to a given set of observations.
(Inherited from UnivariateContinuousDistribution.) |
| Fit(Double, Int32) |
Fits the underlying distribution to a given set of observations.
(Inherited from UnivariateContinuousDistribution.) |
| Fit(Double, Double, IFittingOptions) |
Not supported.
(Overrides UnivariateContinuousDistributionFit(Double, Double, IFittingOptions).) |
| Fit(Double, Int32, IFittingOptions) |
Fits the underlying distribution to a given set of observations.
(Inherited from UnivariateContinuousDistribution.) |
| Generate |
Generates a random observation from the current distribution.
(Inherited from UnivariateContinuousDistribution.) |
| Generate(Int32) |
Generates a random vector of observations from the current distribution.
(Inherited from UnivariateContinuousDistribution.) |
| Generate(Random) |
Generates a random observation from the current distribution.
(Inherited from UnivariateContinuousDistribution.) |
| Generate(Int32, Double) |
Generates a random vector of observations from the current distribution.
(Inherited from UnivariateContinuousDistribution.) |
| Generate(Int32, Random) |
Generates a random vector of observations from the current distribution.
(Inherited from UnivariateContinuousDistribution.) |
| Generate(Int32, Double, Random) |
Generates a random vector of observations from the current distribution.
(Inherited from UnivariateContinuousDistribution.) |
| GetHashCode | Serves as the default hash function. (Inherited from Object.) |
| GetRange |
Gets the distribution range within a given percentile.
(Inherited from UnivariateContinuousDistribution.) |
| GetType | Gets the Type of the current instance. (Inherited from Object.) |
| HazardFunction |
Gets the hazard function, also known as the failure rate or
the conditional failure density function for this distribution
evaluated at point x.
(Inherited from UnivariateContinuousDistribution.) |
| InnerComplementaryDistributionFunction |
Gets the complementary cumulative distribution function
(ccdf) for this distribution evaluated at point x.
This function is also known as the Survival function.
(Inherited from UnivariateContinuousDistribution.) |
| InnerDistributionFunction |
Gets the cumulative distribution function (cdf) for
this distribution evaluated at point x.
(Overrides UnivariateContinuousDistributionInnerDistributionFunction(Double).) |
| InnerInverseDistributionFunction |
Gets the inverse of the cumulative distribution function (icdf) for
this distribution evaluated at probability p. This function
is also known as the Quantile function.
(Overrides UnivariateContinuousDistributionInnerInverseDistributionFunction(Double).) |
| InnerLogProbabilityDensityFunction |
Gets the log-probability density function (pdf) for
this distribution evaluated at point x.
(Overrides UnivariateContinuousDistributionInnerLogProbabilityDensityFunction(Double).) |
| InnerProbabilityDensityFunction |
Gets the probability density function (pdf) for
this distribution evaluated at point x.
(Overrides UnivariateContinuousDistributionInnerProbabilityDensityFunction(Double).) |
| InverseDistributionFunction |
Gets the inverse of the cumulative distribution function (icdf) for
this distribution evaluated at probability p. This function
is also known as the Quantile function.
(Inherited from UnivariateContinuousDistribution.) |
| LogCumulativeHazardFunction |
Gets the log of the cumulative hazard function for this
distribution evaluated at point x.
(Inherited from UnivariateContinuousDistribution.) |
| LogProbabilityDensityFunction |
Gets the log-probability density function (pdf) for
this distribution evaluated at point x.
(Inherited from UnivariateContinuousDistribution.) |
| MemberwiseClone | Creates a shallow copy of the current Object. (Inherited from Object.) |
| ProbabilityDensityFunction |
Gets the probability density function (pdf) for
this distribution evaluated at point x.
(Inherited from UnivariateContinuousDistribution.) |
| QuantileDensityFunction |
Gets the first derivative of the
inverse distribution function (icdf) for this distribution evaluated
at probability p.
(Inherited from UnivariateContinuousDistribution.) |
| ToString |
Returns a String that represents this instance.
(Inherited from DistributionBase.) |
| ToString(IFormatProvider) |
Returns a String that represents this instance.
(Inherited from DistributionBase.) |
| ToString(String) |
Returns a String that represents this instance.
(Inherited from DistributionBase.) |
| ToString(String, IFormatProvider) |
Returns a String that represents this instance.
(Overrides DistributionBaseToString(String, IFormatProvider).) |
Extension Methods| Name | Description | |
|---|---|---|
 | HasMethod |
Checks whether an object implements a method with the given name.
(Defined by ExtensionMethods.) |
| IsEqual |
Compares two objects for equality, performing an elementwise
comparison if the elements are vectors or matrices.
(Defined by Matrix.) |
| To(Type) | Overloaded.
Converts an object into another type, irrespective of whether
the conversion can be done at compile time or not. This can be
used to convert generic types to numeric types during runtime.
(Defined by ExtensionMethods.) |
| ToT | Overloaded.
Converts an object into another type, irrespective of whether
the conversion can be done at compile time or not. This can be
used to convert generic types to numeric types during runtime.
(Defined by ExtensionMethods.) |
RemarksIn probability and statistics, Student's t-distribution (or simply the t-distribution) is a family of continuous probability distributions that arises when estimating the mean of a normally distributed population in situations where the sample size is small and population standard deviation is unknown. It plays a role in a number of widely used statistical analyses, including the Student's t-test for assessing the statistical significance of the difference between two sample means, the construction of confidence intervals for the difference between two population means, and in linear regression analysis. The Student's t-distribution also arises in the Bayesian analysis of data from a normal family.
If we take k samples from a normal distribution with fixed unknown mean and variance, and if we compute the sample mean and sample variance for these k samples, then the t-distribution (for k) can be defined as the distribution of the location of the true mean, relative to the sample mean and divided by the sample standard deviation, after multiplying by the normalizing term sqrt(n), where n is the sample size. In this way the t-distribution can be used to estimate how likely it is that the true mean lies in any given range.
The t-distribution is symmetric and bell-shaped, like the normal distribution, but has heavier tails, meaning that it is more prone to producing values that fall far from its mean. This makes it useful for understanding the statistical behavior of certain types of ratios of random quantities, in which variation in the denominator is amplified and may produce outlying values when the denominator of the ratio falls close to zero. The Student's t-distribution is a special case of the generalized hyperbolic distribution.
References:
Examples// Create a new Student's T distribution with d.f = 4.2 TDistribution t = new TDistribution(degreesOfFreedom: 4.2); // Common measures double mean = t.Mean; // 0.0 double median = t.Median; // 0.0 double var = t.Variance; // 1.9090909090909089 // Cumulative distribution functions double cdf = t.DistributionFunction(x: 1.4); // 0.88456136730659074 double pdf = t.ProbabilityDensityFunction(x: 1.4); // 0.13894002185341031 double lpdf = t.LogProbabilityDensityFunction(x: 1.4); // -1.9737129364307417 // Probability density functions double ccdf = t.ComplementaryDistributionFunction(x: 1.4); // 0.11543863269340926 double icdf = t.InverseDistributionFunction(p: cdf); // 1.4000000000000012 // Hazard (failure rate) functions double hf = t.HazardFunction(x: 1.4); // 1.2035833984833988 double chf = t.CumulativeHazardFunction(x: 1.4); // 2.1590162088918525 // String representation string str = t.ToString(CultureInfo.InvariantCulture); // T(x; df = 4.2)
See Also