NormalDistribution (NMDEV 1.2.9 API)

java.lang.Object
- dev.nm.stat.distribution.univariate.NormalDistribution

All Implemented Interfaces:

ProbabilityDistribution
```
public class NormalDistribution
extends Object
implements ProbabilityDistribution
```
The Normal distribution has its density a Gaussian function. The Normal distribution is probably the most important single distribution. By the central limit theorem, under certain conditions, the sum of a number of random variables with finite means and variances approaches a Normal distribution as the number of variables increases. Laplace proved that the Normal distribution occurs as a limiting distribution of arithmetic means of independent, identically distributed random variables with finite second moment.
The R equivalent functions are dnorm, pnorm, qnorm, rnorm.
Using probabilities and quantiles

The second constructor argument is the standard deviation, not the variance. Supply a finite, strictly positive value; this constructor does not validate it. cdf(x) returns a probability, while quantile(p) takes a probability and returns a value in the original units. density(x) is a density, not the probability of an exact point. kurtosis() returns excess kurtosis (zero for a normal distribution).

Example
```
 import dev.nm.stat.distribution.univariate.NormalDistribution;

 NormalDistribution normal = new NormalDistribution(10.0, 2.0);
 double probability = normal.cdf(10.0); // 0.5
 assert Math.abs(probability - 0.5) < 1e-12;
 assert Math.abs(normal.quantile(0.5) - 10.0) < 1e-12;
 assert normal.variance() == 4.0;
 
```
Entropy and units

Entropy is measured in nats. Multiplying the standard deviation by a positive factor adds the logarithm of that factor.
```
 import dev.nm.stat.distribution.univariate.NormalDistribution;
 NormalDistribution standard = new NormalDistribution();
 NormalDistribution scaled = new NormalDistribution(20, 3);
 assert Math.abs(standard.entropy() - 1.4189385332046727) < 1e-12;
 assert Math.abs(scaled.entropy() - standard.entropy() - Math.log(3)) < 1e-12;
 
```
See Also:
- Wikipedia: Normal distribution
- Gaussian

Constructor Summary

Constructors
Constructor and Description
`NormalDistribution()` Construct an instance of the standard Normal distribution with mean 0 and standard deviation 1.
`NormalDistribution(double mu, double sigma)` Construct a Normal distribution with mean `mu` and standard deviation `sigma`.

Method Summary

All Methods Instance Methods Concrete Methods
Modifier and Type	Method and Description
`double`	`cdf(double x)` Gets the cumulative probability F(x) = Pr(X ≤ x).
`double`	`density(double x)` The density function, which, if exists, is the derivative of F.
`double`	`entropy()` Returns differential entropy in nats (natural-logarithm units).
`double`	`kurtosis()` Gets the excess kurtosis of this distribution.
`double`	`mean()` Gets the mean of this distribution.
`double`	`median()` Gets the median of this distribution.
`double`	`moment(double t)` The moment generating function is the expected value of \(e^{tX}\).
`double`	`quantile(double u)` Gets the quantile, the inverse of the cumulative distribution function.
`double`	`skew()` Gets the skewness of this distribution.
`double`	`variance()` Gets the variance of this distribution.

Methods inherited from class java.lang.Object
clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

- Constructor Detail
  - NormalDistribution
```
public NormalDistribution()
```
    Construct an instance of the standard Normal distribution with mean 0 and standard deviation 1.
  - NormalDistribution
```
public NormalDistribution(double mu,
                          double sigma)
```
    Construct a Normal distribution with mean mu and standard deviation sigma.
    
    Parameters:
    
    mu - the mean
    
    sigma - the standard deviation; the caller must supply a finite value greater than zero
- Method Detail
  - mean
```
public double mean()
```
    Description copied from interface: ProbabilityDistribution
    
    Gets the mean of this distribution.
    
    Specified by:
    
    mean in interface ProbabilityDistribution
    
    Returns:
    
    the mean
    
    See Also:
    
    Wikipedia: Expected value
  - median
```
public double median()
```
    Description copied from interface: ProbabilityDistribution
    
    Gets the median of this distribution.
    
    Specified by:
    
    median in interface ProbabilityDistribution
    
    Returns:
    
    the median
    
    See Also:
    
    Wikipedia: Median
  - variance
```
public double variance()
```
    Description copied from interface: ProbabilityDistribution
    
    Gets the variance of this distribution.
    
    Specified by:
    
    variance in interface ProbabilityDistribution
    
    Returns:
    
    the variance
    
    See Also:
    
    Wikipedia: Variance
  - skew
```
public double skew()
```
    Description copied from interface: ProbabilityDistribution
    
    Gets the skewness of this distribution.
    
    Specified by:
    
    skew in interface ProbabilityDistribution
    
    Returns:
    
    the skewness
    
    See Also:
    
    Wikipedia: Skewness
  - kurtosis
```
public double kurtosis()
```
    Description copied from interface: ProbabilityDistribution
    
    Gets the excess kurtosis of this distribution.
    
    Specified by:
    
    kurtosis in interface ProbabilityDistribution
    
    Returns:
    
    the excess kurtosis
    
    See Also:
    
    Wikipedia: Kurtosis
  - entropy
```
public double entropy()
```
    Returns differential entropy in nats (natural-logarithm units). The value is \(\frac{1}{2}\log(2\pi e)+\log\sigma\). Computing the logarithm of the scale separately avoids overflow or underflow from squaring an otherwise valid finite scale.
    
    Specified by:
    
    entropy in interface ProbabilityDistribution
    
    Returns:
    
    the normal differential entropy in nats
    
    See Also:
    
    Wikipedia: Entropy (information theory)
  - cdf
```
public double cdf(double x)
```
    Description copied from interface: ProbabilityDistribution
    
    Gets the cumulative probability F(x) = Pr(X ≤ x).
    
    Specified by:
    
    cdf in interface ProbabilityDistribution
    
    Parameters:
    
    x - x
    
    Returns:
    
    F(x) = Pr(X ≤ x)
    
    See Also:
    
    Wikipedia: Cumulative distribution function
  - quantile
```
public double quantile(double u)
```
    Description copied from interface: ProbabilityDistribution
    
    Gets the quantile, the inverse of the cumulative distribution function. It is the value below which random draws from the distribution would fall u×100 percent of the time. \[ \begin{gathered} F^{-1}(u) = x, such that \\ Pr(X \le x) = u \end{gathered} \] This may not always exist.
    
    Specified by:
    
    quantile in interface ProbabilityDistribution
    
    Parameters:
    
    u - u, a quantile
    
    Returns:
    
    \(F^{-1}(u)\)
    
    See Also:
    
    Wikipedia: Quantile function
  - density
```
public double density(double x)
```
    Description copied from interface: ProbabilityDistribution
    
    The density function, which, if exists, is the derivative of F. It describes the density of probability at each point in the sample space. \[ f(x) = dF(X) / dx \] This may not always exist.
    For the discrete cases, this is the probability mass function. It gives the probability that a discrete random variable is exactly equal to some value.
    Specified by:
    
    density in interface ProbabilityDistribution
    
    Parameters:
    
    x - x
    
    Returns:
    
    f(x)
    
    See Also:
    Wikipedia: Probability density function
    
    Wikipedia: Probability mass function
  - moment
```
public double moment(double t)
```
    Description copied from interface: ProbabilityDistribution
    
    The moment generating function is the expected value of \(e^{tX}\). That is, \[ E(e^{tX}) \] This may not always exist.
    
    Specified by:
    
    moment in interface ProbabilityDistribution
    
    Parameters:
    
    t - t
    
    Returns:
    
    E(exp(tX))
    
    See Also:
    
    Wikipedia: Moment-generating function

Class NormalDistribution

Using probabilities and quantiles

Example

Entropy and units

Constructor Summary

Method Summary

Methods inherited from class java.lang.Object

Constructor Detail

NormalDistribution

NormalDistribution

Method Detail

mean

median

variance

skew

kurtosis

entropy

cdf

quantile

density

moment