SSJ
V. 2.6.

## umontreal.iro.lecuyer.probdist Class NormalDist

```java.lang.Object
umontreal.iro.lecuyer.probdist.ContinuousDistribution
umontreal.iro.lecuyer.probdist.NormalDist
```
All Implemented Interfaces:
Distribution
Direct Known Subclasses:
NormalDistQuick

`public class NormalDistextends ContinuousDistribution`

Extends the class `ContinuousDistribution` for the normal distribution (e.g.,). It has mean μ and variance σ2. Its density function is

f (x) = e-(x-μ)2/(2σ2)/((2π)1/2σ)        for - ∞ < x < ∞,

where σ > 0. When μ = 0 and σ = 1, we have the standard normal distribution, with corresponding distribution function

F(x) = Φ(x) = ∫-∞xe-t2/2 dt/(2π)1/2        for - ∞ < x < ∞.

The non-static methods cdf, barF, and inverseF are implemented via `cdf01`, `barF01`, and `inverseF01`, respectively.

Field Summary

Fields inherited from class umontreal.iro.lecuyer.probdist.ContinuousDistribution
`decPrec`

Constructor Summary
`NormalDist()`
Constructs a NormalDist object with default parameters μ = 0 and σ = 1.
```NormalDist(double mu, double sigma)```
Constructs a NormalDist object with mean μ = mu and standard deviation σ = sigma.

Method Summary
` double` `barF(double x)`
Returns the complementary distribution function.
`static double` ```barF(double mu, double sigma, double x)```
Computes the complementary normal distribution function bar(F)(x) = 1 - Φ((x - μ)/σ), with mean μ and variance σ2.
`static double` `barF01(double x)`
Same as `barF` (0, 1, x).
` double` `cdf(double x)`
Returns the distribution function F(x).
`static double` ```cdf(double mu, double sigma, double x)```
Computes the normal distribution function with mean μ and variance σ2.
`static double` `cdf01(double x)`
Same as `cdf` (0, 1, x).
` double` `density(double x)`
Returns f (x), the density evaluated at x.
`static double` ```density(double mu, double sigma, double x)```
Computes the normal density function.
`static double` `density01(double x)`
Same as `density` (0, 1, x).
`static NormalDist` ```getInstanceFromMLE(double[] x, int n)```
Creates a new instance of a normal distribution with parameters μ and σ estimated using the maximum likelihood method based on the n observations x[i], i = 0, 1,…, n - 1.
` double` `getMean()`
Returns the mean.
`static double` ```getMean(double mu, double sigma)```
Computes and returns the mean E[X] = μ of the normal distribution with parameters μ and σ.
`static double[]` ```getMLE(double[] x, int n)```
Estimates the parameters (μ, σ) of the normal distribution using the maximum likelihood method, from the n observations x[i], i = 0, 1,…, n - 1.
` double` `getMu()`
Returns the parameter μ.
` double[]` `getParams()`
Return a table containing the parameters of the current distribution.
` double` `getSigma()`
Returns the parameter σ.
` double` `getStandardDeviation()`
Returns the standard deviation.
`static double` ```getStandardDeviation(double mu, double sigma)```
Computes and returns the standard deviation σ of the normal distribution with parameters μ and σ.
` double` `getVariance()`
Returns the variance.
`static double` ```getVariance(double mu, double sigma)```
Computes and returns the variance Var[X] = σ2 of the normal distribution with parameters μ and σ.
` double` `inverseF(double u)`
Returns the inverse distribution function x = F-1(u).
`static double` ```inverseF(double mu, double sigma, double u)```
Computes the inverse normal distribution function with mean μ and variance σ2.
`static double` `inverseF01(double u)`
Same as `inverseF` (0, 1, u).
` void` ```setParams(double mu, double sigma)```
Sets the parameters μ and σ of this object.
` String` `toString()`

Methods inherited from class umontreal.iro.lecuyer.probdist.ContinuousDistribution
`getXinf, getXsup, inverseBisection, inverseBrent, setXinf, setXsup`

Methods inherited from class java.lang.Object
`equals, getClass, hashCode, notify, notifyAll, wait, wait, wait`

Constructor Detail

### NormalDist

`public NormalDist()`
Constructs a NormalDist object with default parameters μ = 0 and σ = 1.

### NormalDist

```public NormalDist(double mu,
double sigma)```
Constructs a NormalDist object with mean μ = mu and standard deviation σ = sigma.

Method Detail

### density

`public double density(double x)`
Description copied from class: `ContinuousDistribution`
Returns f (x), the density evaluated at x.

Specified by:
`density` in class `ContinuousDistribution`
Parameters:
`x` - value at which the density is evaluated
Returns:
density function evaluated at x

### cdf

`public double cdf(double x)`
Description copied from interface: `Distribution`
Returns the distribution function F(x).

Parameters:
`x` - value at which the distribution function is evaluated
Returns:
distribution function evaluated at x

### barF

`public double barF(double x)`
Description copied from class: `ContinuousDistribution`
Returns the complementary distribution function. The default implementation computes bar(F)(x) = 1 - F(x).

Specified by:
`barF` in interface `Distribution`
Overrides:
`barF` in class `ContinuousDistribution`
Parameters:
`x` - value at which the complementary distribution function is evaluated
Returns:
complementary distribution function evaluated at x

### inverseF

`public double inverseF(double u)`
Description copied from class: `ContinuousDistribution`
Returns the inverse distribution function x = F-1(u). Restrictions: u∈[0, 1].

Specified by:
`inverseF` in interface `Distribution`
Overrides:
`inverseF` in class `ContinuousDistribution`
Parameters:
`u` - value at which the inverse distribution function is evaluated
Returns:
the inverse distribution function evaluated at u

### getMean

`public double getMean()`
Description copied from class: `ContinuousDistribution`
Returns the mean.

Specified by:
`getMean` in interface `Distribution`
Overrides:
`getMean` in class `ContinuousDistribution`
Returns:
the mean

### getVariance

`public double getVariance()`
Description copied from class: `ContinuousDistribution`
Returns the variance.

Specified by:
`getVariance` in interface `Distribution`
Overrides:
`getVariance` in class `ContinuousDistribution`
Returns:
the variance

### getStandardDeviation

`public double getStandardDeviation()`
Description copied from class: `ContinuousDistribution`
Returns the standard deviation.

Specified by:
`getStandardDeviation` in interface `Distribution`
Overrides:
`getStandardDeviation` in class `ContinuousDistribution`
Returns:
the standard deviation

### density01

`public static double density01(double x)`
Same as `density` (0, 1, x).

### density

```public static double density(double mu,
double sigma,
double x)```
Computes the normal density function.

### cdf01

`public static double cdf01(double x)`
Same as `cdf` (0, 1, x).

### cdf

```public static double cdf(double mu,
double sigma,
double x)```
Computes the normal distribution function with mean μ and variance σ2. Uses the Chebyshev approximation , which gives 16 decimals of precision.

### barF01

`public static double barF01(double x)`
Same as `barF` (0, 1, x).

### barF

```public static double barF(double mu,
double sigma,
double x)```
Computes the complementary normal distribution function bar(F)(x) = 1 - Φ((x - μ)/σ), with mean μ and variance σ2. Uses a Chebyshev series giving 16 decimal digits of precision.

### inverseF01

`public static double inverseF01(double u)`
Same as `inverseF` (0, 1, u).

### inverseF

```public static double inverseF(double mu,
double sigma,
double u)```
Computes the inverse normal distribution function with mean μ and variance σ2. Uses different rational Chebyshev approximations. Returns 16 decimal digits of precision for 2.2×10-308 < u < 1.

### getMLE

```public static double[] getMLE(double[] x,
int n)```
Estimates the parameters (μ, σ) of the normal distribution using the maximum likelihood method, from the n observations x[i], i = 0, 1,…, n - 1. The estimates are returned in a two-element array, in regular order: [hat(μ), hat(σ)].

Parameters:
`x` - the list of observations used to evaluate parameters
`n` - the number of observations used to evaluate parameters
Returns:
returns the parameters [hat(μ), hat(σ)]

### getInstanceFromMLE

```public static NormalDist getInstanceFromMLE(double[] x,
int n)```
Creates a new instance of a normal distribution with parameters μ and σ estimated using the maximum likelihood method based on the n observations x[i], i = 0, 1,…, n - 1.

Parameters:
`x` - the list of observations to use to evaluate parameters
`n` - the number of observations to use to evaluate parameters

### getMean

```public static double getMean(double mu,
double sigma)```
Computes and returns the mean E[X] = μ of the normal distribution with parameters μ and σ.

Returns:
the mean of the normal distribution E[X] = μ

### getVariance

```public static double getVariance(double mu,
double sigma)```
Computes and returns the variance Var[X] = σ2 of the normal distribution with parameters μ and σ.

Returns:
the variance of the normal distribution Var[X] = σ2

### getStandardDeviation

```public static double getStandardDeviation(double mu,
double sigma)```
Computes and returns the standard deviation σ of the normal distribution with parameters μ and σ.

Returns:
the standard deviation of the normal distribution

### getMu

`public double getMu()`
Returns the parameter μ.

### getSigma

`public double getSigma()`
Returns the parameter σ.

### setParams

```public void setParams(double mu,
double sigma)```
Sets the parameters μ and σ of this object.

### getParams

`public double[] getParams()`
Return a table containing the parameters of the current distribution. This table is put in regular order: [μ, σ].

### toString

`public String toString()`
Overrides:
`toString` in class `Object`

SSJ
V. 2.6.

To submit a bug or ask questions, send an e-mail to Pierre L'Ecuyer.