## org.apache.commons.math3.distribution Class NormalDistribution

```java.lang.Object org.apache.commons.math3.distribution.AbstractRealDistribution org.apache.commons.math3.distribution.NormalDistribution
```
All Implemented Interfaces:
Serializable, RealDistribution

`public class NormalDistributionextends AbstractRealDistribution`

Implementation of the normal (gaussian) distribution.

Version:
\$Id: NormalDistribution.java 1462423 2013-03-29 07:25:18Z luc \$
Normal distribution (Wikipedia), Normal distribution (MathWorld), Serialized Form

Field Summary
`static double` `DEFAULT_INVERSE_ABSOLUTE_ACCURACY`
Default inverse cumulative probability accuracy.

Fields inherited from class org.apache.commons.math3.distribution.AbstractRealDistribution
`random, randomData, SOLVER_DEFAULT_ABSOLUTE_ACCURACY`

Constructor Summary
`NormalDistribution()`
Create a normal distribution with mean equal to zero and standard deviation equal to one.
```NormalDistribution(double mean, double sd)```
Create a normal distribution using the given mean and standard deviation.
```NormalDistribution(double mean, double sd, double inverseCumAccuracy)```
Create a normal distribution using the given mean, standard deviation and inverse cumulative distribution accuracy.
```NormalDistribution(RandomGenerator rng, double mean, double sd, double inverseCumAccuracy)```
Creates a normal distribution.

Method Summary
` double` `cumulativeProbability(double x)`
For a random variable `X` whose values are distributed according to this distribution, this method returns `P(X <= x)`.
` double` ```cumulativeProbability(double x0, double x1)```
Deprecated. See `RealDistribution.cumulativeProbability(double,double)`
` double` `density(double x)`
Returns the probability density function (PDF) of this distribution evaluated at the specified point `x`.
` double` `getMean()`
Access the mean.
` double` `getNumericalMean()`
Use this method to get the numerical value of the mean of this distribution.
` double` `getNumericalVariance()`
Use this method to get the numerical value of the variance of this distribution.
`protected  double` `getSolverAbsoluteAccuracy()`
Returns the solver absolute accuracy for inverse cumulative computation.
` double` `getStandardDeviation()`
Access the standard deviation.
` double` `getSupportLowerBound()`
Access the lower bound of the support.
` double` `getSupportUpperBound()`
Access the upper bound of the support.
` double` `inverseCumulativeProbability(double p)`
Computes the quantile function of this distribution.
` boolean` `isSupportConnected()`
Use this method to get information about whether the support is connected, i.e. whether all values between the lower and upper bound of the support are included in the support.
` boolean` `isSupportLowerBoundInclusive()`
Whether or not the lower bound of support is in the domain of the density function.
` boolean` `isSupportUpperBoundInclusive()`
Whether or not the upper bound of support is in the domain of the density function.
` double` ```probability(double x0, double x1)```
For a random variable `X` whose values are distributed according to this distribution, this method returns `P(x0 < X <= x1)`.
` double` `sample()`
Generate a random value sampled from this distribution.

Methods inherited from class org.apache.commons.math3.distribution.AbstractRealDistribution
`probability, reseedRandomGenerator, sample`

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

Field Detail

### DEFAULT_INVERSE_ABSOLUTE_ACCURACY

`public static final double DEFAULT_INVERSE_ABSOLUTE_ACCURACY`
Default inverse cumulative probability accuracy.

Since:
2.1
Constant Field Values
Constructor Detail

### NormalDistribution

`public NormalDistribution()`
Create a normal distribution with mean equal to zero and standard deviation equal to one.

### NormalDistribution

```public NormalDistribution(double mean,
double sd)
throws NotStrictlyPositiveException```
Create a normal distribution using the given mean and standard deviation.

Parameters:
`mean` - Mean for this distribution.
`sd` - Standard deviation for this distribution.
Throws:
`NotStrictlyPositiveException` - if `sd <= 0`.

### NormalDistribution

```public NormalDistribution(double mean,
double sd,
double inverseCumAccuracy)
throws NotStrictlyPositiveException```
Create a normal distribution using the given mean, standard deviation and inverse cumulative distribution accuracy.

Parameters:
`mean` - Mean for this distribution.
`sd` - Standard deviation for this distribution.
`inverseCumAccuracy` - Inverse cumulative probability accuracy.
Throws:
`NotStrictlyPositiveException` - if `sd <= 0`.
Since:
2.1

### NormalDistribution

```public NormalDistribution(RandomGenerator rng,
double mean,
double sd,
double inverseCumAccuracy)
throws NotStrictlyPositiveException```
Creates a normal distribution.

Parameters:
`rng` - Random number generator.
`mean` - Mean for this distribution.
`sd` - Standard deviation for this distribution.
`inverseCumAccuracy` - Inverse cumulative probability accuracy.
Throws:
`NotStrictlyPositiveException` - if `sd <= 0`.
Since:
3.1
Method Detail

### getMean

`public double getMean()`
Access the mean.

Returns:
the mean for this distribution.

### getStandardDeviation

`public double getStandardDeviation()`
Access the standard deviation.

Returns:
the standard deviation for this distribution.

### density

`public double density(double x)`
Returns the probability density function (PDF) of this distribution evaluated at the specified point `x`. In general, the PDF is the derivative of the `CDF`. If the derivative does not exist at `x`, then an appropriate replacement should be returned, e.g. `Double.POSITIVE_INFINITY`, `Double.NaN`, or the limit inferior or limit superior of the difference quotient.

Parameters:
`x` - the point at which the PDF is evaluated
Returns:
the value of the probability density function at point `x`

### cumulativeProbability

`public double cumulativeProbability(double x)`
For a random variable `X` whose values are distributed according to this distribution, this method returns `P(X <= x)`. In other words, this method represents the (cumulative) distribution function (CDF) for this distribution. If `x` is more than 40 standard deviations from the mean, 0 or 1 is returned, as in these cases the actual value is within `Double.MIN_VALUE` of 0 or 1.

Parameters:
`x` - the point at which the CDF is evaluated
Returns:
the probability that a random variable with this distribution takes a value less than or equal to `x`

### inverseCumulativeProbability

```public double inverseCumulativeProbability(double p)
throws OutOfRangeException```
Computes the quantile function of this distribution. For a random variable `X` distributed according to this distribution, the returned value is
• `inf{x in R | P(X<=x) >= p}` for `0 < p <= 1`,
• `inf{x in R | P(X<=x) > 0}` for `p = 0`.
The default implementation returns

Specified by:
`inverseCumulativeProbability` in interface `RealDistribution`
Overrides:
`inverseCumulativeProbability` in class `AbstractRealDistribution`
Parameters:
`p` - the cumulative probability
Returns:
the smallest `p`-quantile of this distribution (largest 0-quantile for `p = 0`)
Throws:
`OutOfRangeException` - if `p < 0` or `p > 1`
Since:
3.2

### cumulativeProbability

```@Deprecated
public double cumulativeProbability(double x0,
double x1)
throws NumberIsTooLargeException```
Deprecated. See `RealDistribution.cumulativeProbability(double,double)`

For a random variable `X` whose values are distributed according to this distribution, this method returns `P(x0 < X <= x1)`. The default implementation uses the identity

`P(x0 < X <= x1) = P(X <= x1) - P(X <= x0)`

Specified by:
`cumulativeProbability` in interface `RealDistribution`
Overrides:
`cumulativeProbability` in class `AbstractRealDistribution`
Parameters:
`x0` - the exclusive lower bound
`x1` - the inclusive upper bound
Returns:
the probability that a random variable with this distribution takes a value between `x0` and `x1`, excluding the lower and including the upper endpoint
Throws:
`NumberIsTooLargeException` - if `x0 > x1`

### probability

```public double probability(double x0,
double x1)
throws NumberIsTooLargeException```
For a random variable `X` whose values are distributed according to this distribution, this method returns `P(x0 < X <= x1)`.

Overrides:
`probability` in class `AbstractRealDistribution`
Parameters:
`x0` - Lower bound (excluded).
`x1` - Upper bound (included).
Returns:
the probability that a random variable with this distribution takes a value between `x0` and `x1`, excluding the lower and including the upper endpoint.
Throws:
`NumberIsTooLargeException` - if `x0 > x1`. The default implementation uses the identity `P(x0 < X <= x1) = P(X <= x1) - P(X <= x0)`

### getSolverAbsoluteAccuracy

`protected double getSolverAbsoluteAccuracy()`
Returns the solver absolute accuracy for inverse cumulative computation. You can override this method in order to use a Brent solver with an absolute accuracy different from the default.

Overrides:
`getSolverAbsoluteAccuracy` in class `AbstractRealDistribution`
Returns:
the maximum absolute error in inverse cumulative probability estimates

### getNumericalMean

`public double getNumericalMean()`
Use this method to get the numerical value of the mean of this distribution. For mean parameter `mu`, the mean is `mu`.

Returns:
the mean or `Double.NaN` if it is not defined

### getNumericalVariance

`public double getNumericalVariance()`
Use this method to get the numerical value of the variance of this distribution. For standard deviation parameter `s`, the variance is `s^2`.

Returns:
the variance (possibly `Double.POSITIVE_INFINITY` as for certain cases in `TDistribution`) or `Double.NaN` if it is not defined

### getSupportLowerBound

`public double getSupportLowerBound()`
Access the lower bound of the support. This method must return the same value as `inverseCumulativeProbability(0)`. In other words, this method must return

`inf {x in R | P(X <= x) > 0}`.

The lower bound of the support is always negative infinity no matter the parameters.

Returns:
lower bound of the support (always `Double.NEGATIVE_INFINITY`)

### getSupportUpperBound

`public double getSupportUpperBound()`
Access the upper bound of the support. This method must return the same value as `inverseCumulativeProbability(1)`. In other words, this method must return

`inf {x in R | P(X <= x) = 1}`.

The upper bound of the support is always positive infinity no matter the parameters.

Returns:
upper bound of the support (always `Double.POSITIVE_INFINITY`)

### isSupportLowerBoundInclusive

`public boolean isSupportLowerBoundInclusive()`
Whether or not the lower bound of support is in the domain of the density function. Returns true iff `getSupporLowerBound()` is finite and `density(getSupportLowerBound())` returns a non-NaN, non-infinite value.

Returns:
true if the lower bound of support is finite and the density function returns a non-NaN, non-infinite value there

### isSupportUpperBoundInclusive

`public boolean isSupportUpperBoundInclusive()`
Whether or not the upper bound of support is in the domain of the density function. Returns true iff `getSupportUpperBound()` is finite and `density(getSupportUpperBound())` returns a non-NaN, non-infinite value.

Returns:
true if the upper bound of support is finite and the density function returns a non-NaN, non-infinite value there

### isSupportConnected

`public boolean isSupportConnected()`
Use this method to get information about whether the support is connected, i.e. whether all values between the lower and upper bound of the support are included in the support. The support of this distribution is connected.

Returns:
`true`

### sample

`public double sample()`
Generate a random value sampled from this distribution. The default implementation uses the inversion method.

Specified by:
`sample` in interface `RealDistribution`
Overrides:
`sample` in class `AbstractRealDistribution`
Returns:
a random value.