smile.stat.distribution

## Class ShiftedGeometricDistribution

• All Implemented Interfaces:
java.io.Serializable, DiscreteExponentialFamily, Distribution

```public class ShiftedGeometricDistribution
extends DiscreteDistribution
implements DiscreteExponentialFamily```
The "shifted" geometric distribution is a discrete probability distribution of the number of failures before the first success, supported on the set `{0, 1, 2, 3, …}`. If the probability of success on each trial is p, then the probability that the k-th trial (out of k trials) is the first success is

```     Pr(X = k) = (1 - p)k p
```
.
`GeometricDistribution`, Serialized Form
• ### Field Summary

Fields
Modifier and Type Field and Description
`double` `p`
The probability of success.
• ### Constructor Summary

Constructors
Constructor and Description
`ShiftedGeometricDistribution(double p)`
Constructor.
• ### Method Summary

All Methods
Modifier and Type Method and Description
`double` `cdf(double k)`
Cumulative distribution function.
`double` `entropy()`
Shannon entropy of the distribution.
`static ShiftedGeometricDistribution` `fit(int[] data)`
Estimates the distribution parameters by MLE.
`int` `length()`
The number of parameters of the distribution.
`double` `logp(int k)`
The probability mass function in log scale.
`DiscreteMixture.Component` ```M(int[] x, double[] posteriori)```
The M step in the EM algorithm, which depends the specific distribution.
`double` `mean()`
The mean of distribution.
`double` `p(int k)`
The probability mass function.
`double` `quantile(double p)`
The quantile, the probability to the left of quantile is p.
`double` `rand()`
Generates a random number following this distribution.
`double` `sd()`
The standard deviation of distribution.
`java.lang.String` `toString()`
`double` `variance()`
The variance of distribution.
• ### Methods inherited from class smile.stat.distribution.DiscreteDistribution

`likelihood, logLikelihood, logp, p, quantile, randi, randi`
• ### Methods inherited from class smile.stat.distribution.AbstractDistribution

`inverseTransformSampling, quantile, quantile, rejection`
• ### Methods inherited from class java.lang.Object

`clone, equals, finalize, getClass, hashCode, notify, notifyAll, wait, wait, wait`
• ### Methods inherited from interface smile.stat.distribution.Distribution

`likelihood, logLikelihood, rand`
• ### Field Detail

• #### p

`public final double p`
The probability of success.
• ### Constructor Detail

• #### ShiftedGeometricDistribution

`public ShiftedGeometricDistribution(double p)`
Constructor.
Parameters:
`p` - the probability of success.
• ### Method Detail

• #### fit

`public static ShiftedGeometricDistribution fit(int[] data)`
Estimates the distribution parameters by MLE.
• #### length

`public int length()`
Description copied from interface: `Distribution`
The number of parameters of the distribution. The "length" is in the sense of the minimum description length principle.
Specified by:
`length` in interface `Distribution`
• #### mean

`public double mean()`
Description copied from interface: `Distribution`
The mean of distribution.
Specified by:
`mean` in interface `Distribution`
• #### variance

`public double variance()`
Description copied from interface: `Distribution`
The variance of distribution.
Specified by:
`variance` in interface `Distribution`
• #### sd

`public double sd()`
Description copied from interface: `Distribution`
The standard deviation of distribution.
Specified by:
`sd` in interface `Distribution`
• #### entropy

`public double entropy()`
Description copied from interface: `Distribution`
Shannon entropy of the distribution.
Specified by:
`entropy` in interface `Distribution`
• #### toString

`public java.lang.String toString()`
Overrides:
`toString` in class `java.lang.Object`
• #### rand

`public double rand()`
Description copied from interface: `Distribution`
Generates a random number following this distribution.
Specified by:
`rand` in interface `Distribution`
• #### p

`public double p(int k)`
Description copied from class: `DiscreteDistribution`
The probability mass function.
Specified by:
`p` in class `DiscreteDistribution`
• #### logp

`public double logp(int k)`
Description copied from class: `DiscreteDistribution`
The probability mass function in log scale.
Specified by:
`logp` in class `DiscreteDistribution`
• #### cdf

`public double cdf(double k)`
Description copied from interface: `Distribution`
Cumulative distribution function. That is the probability to the left of x.
Specified by:
`cdf` in interface `Distribution`
• #### quantile

`public double quantile(double p)`
Description copied from interface: `Distribution`
The quantile, the probability to the left of quantile is p. It is actually the inverse of cdf.
Specified by:
`quantile` in interface `Distribution`
• #### M

```public DiscreteMixture.Component M(int[] x,
double[] posteriori)```
Description copied from interface: `DiscreteExponentialFamily`
The M step in the EM algorithm, which depends the specific distribution.
Specified by:
`M` in interface `DiscreteExponentialFamily`
Parameters:
`x` - the input data for estimation
`posteriori` - the posteriori probability.
Returns:
the (unnormalized) weight of this distribution in the mixture.