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 * Licensed to the Apache Software Foundation (ASF) under one or more
 * contributor license agreements.  See the NOTICE file distributed with
 * this work for additional information regarding copyright ownership.
 * The ASF licenses this file to You under the Apache License, Version 2.0
 * (the "License"); you may not use this file except in compliance with
 * the License.  You may obtain a copy of the License at
 *
 *      http://www.apache.org/licenses/LICENSE-2.0
 *
 * Unless required by applicable law or agreed to in writing, software
 * distributed under the License is distributed on an "AS IS" BASIS,
 * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
 * See the License for the specific language governing permissions and
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package org.apache.commons.math3.distribution;

import org.apache.commons.math3.exception.NotStrictlyPositiveException;
import org.apache.commons.math3.exception.NumberIsTooLargeException;
import org.apache.commons.math3.exception.OutOfRangeException;
import org.apache.commons.math3.exception.util.LocalizedFormats;
import org.apache.commons.math3.random.RandomGenerator;
import org.apache.commons.math3.random.Well19937c;
import org.apache.commons.math3.special.Erf;
import org.apache.commons.math3.util.FastMath;

Implementation of the normal (gaussian) distribution.
See Also:
Normal distribution (Wikipedia)
Normal distribution (MathWorld)/**
 * Implementation of the normal (gaussian) distribution.
 *
 * @see <a href="http://en.wikipedia.org/wiki/Normal_distribution">Normal distribution (Wikipedia)</a>
 * @see <a href="http://mathworld.wolfram.com/NormalDistribution.html">Normal distribution (MathWorld)</a>
 */
public class NormalDistribution extends AbstractRealDistribution {
    
Default inverse cumulative probability accuracy.
Since:
2.1/**
     * Default inverse cumulative probability accuracy.
     * @since 2.1
     */
    public static final double DEFAULT_INVERSE_ABSOLUTE_ACCURACY = 1e-9;
    
Serializable version identifier. /** Serializable version identifier. */
    private static final long serialVersionUID = 8589540077390120676L;
    
√(2) /** &radic;(2) */
    private static final double SQRT2 = FastMath.sqrt(2.0);
    
Mean of this distribution. /** Mean of this distribution. */
    private final double mean;
    
Standard deviation of this distribution. /** Standard deviation of this distribution. */
    private final double standardDeviation;
    
The value of log(sd) + 0.5*log(2*pi) stored for faster computation. /** The value of {@code log(sd) + 0.5*log(2*pi)} stored for faster computation. */
    private final double logStandardDeviationPlusHalfLog2Pi;
    
Inverse cumulative probability accuracy. /** Inverse cumulative probability accuracy. */
    private final double solverAbsoluteAccuracy;

    
Create a normal distribution with mean equal to zero and standard
deviation equal to one.

Note: this constructor will implicitly create an instance of Well19937c as random generator to be used for sampling only (see sample() and AbstractRealDistribution.sample(int)). In case no sampling is needed for the created distribution, it is advised to pass null as random generator via the appropriate constructors to avoid the additional initialisation overhead. /**
     * Create a normal distribution with mean equal to zero and standard
     * deviation equal to one.
     * <p>
     * <b>Note:</b> this constructor will implicitly create an instance of
     * {@link Well19937c} as random generator to be used for sampling only (see
     * {@link #sample()} and {@link #sample(int)}). In case no sampling is
     * needed for the created distribution, it is advised to pass {@code null}
     * as random generator via the appropriate constructors to avoid the
     * additional initialisation overhead.
     */
    public NormalDistribution() {
        this(0, 1);
    }

    
Create a normal distribution using the given mean and standard deviation.

Note: this constructor will implicitly create an instance of Well19937c as random generator to be used for sampling only (see sample() and AbstractRealDistribution.sample(int)). In case no sampling is needed for the created distribution, it is advised to pass null as random generator via the appropriate constructors to avoid the additional initialisation overhead. 
Params:
mean – Mean for this distribution.
sd – Standard deviation for this distribution.
Throws:
NotStrictlyPositiveException – if sd <= 0./**
     * Create a normal distribution using the given mean and standard deviation.
     * <p>
     * <b>Note:</b> this constructor will implicitly create an instance of
     * {@link Well19937c} as random generator to be used for sampling only (see
     * {@link #sample()} and {@link #sample(int)}). In case no sampling is
     * needed for the created distribution, it is advised to pass {@code null}
     * as random generator via the appropriate constructors to avoid the
     * additional initialisation overhead.
     *
     * @param mean Mean for this distribution.
     * @param sd Standard deviation for this distribution.
     * @throws NotStrictlyPositiveException if {@code sd <= 0}.
     */
    public NormalDistribution(double mean, double sd)
        throws NotStrictlyPositiveException {
        this(mean, sd, DEFAULT_INVERSE_ABSOLUTE_ACCURACY);
    }

    
Create a normal distribution using the given mean, standard deviation and
inverse cumulative distribution accuracy.

Note: this constructor will implicitly create an instance of Well19937c as random generator to be used for sampling only (see sample() and AbstractRealDistribution.sample(int)). In case no sampling is needed for the created distribution, it is advised to pass null as random generator via the appropriate constructors to avoid the additional initialisation overhead. 
Params:
mean – Mean for this distribution.
sd – Standard deviation for this distribution.
inverseCumAccuracy – Inverse cumulative probability accuracy.
Throws:
NotStrictlyPositiveException – if sd <= 0.
Since:
2.1/**
     * Create a normal distribution using the given mean, standard deviation and
     * inverse cumulative distribution accuracy.
     * <p>
     * <b>Note:</b> this constructor will implicitly create an instance of
     * {@link Well19937c} as random generator to be used for sampling only (see
     * {@link #sample()} and {@link #sample(int)}). In case no sampling is
     * needed for the created distribution, it is advised to pass {@code null}
     * as random generator via the appropriate constructors to avoid the
     * additional initialisation overhead.
     *
     * @param mean Mean for this distribution.
     * @param sd Standard deviation for this distribution.
     * @param inverseCumAccuracy Inverse cumulative probability accuracy.
     * @throws NotStrictlyPositiveException if {@code sd <= 0}.
     * @since 2.1
     */
    public NormalDistribution(double mean, double sd, double inverseCumAccuracy)
        throws NotStrictlyPositiveException {
        this(new Well19937c(), mean, sd, inverseCumAccuracy);
    }

    
Creates a normal distribution.
Params:
rng – Random number generator.
mean – Mean for this distribution.
sd – Standard deviation for this distribution.
Throws:
NotStrictlyPositiveException – if sd <= 0.
Since:
3.3/**
     * Creates a normal distribution.
     *
     * @param rng Random number generator.
     * @param mean Mean for this distribution.
     * @param sd Standard deviation for this distribution.
     * @throws NotStrictlyPositiveException if {@code sd <= 0}.
     * @since 3.3
     */
    public NormalDistribution(RandomGenerator rng, double mean, double sd)
        throws NotStrictlyPositiveException {
        this(rng, mean, sd, DEFAULT_INVERSE_ABSOLUTE_ACCURACY);
    }

    
Creates a normal distribution.
Params:
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/**
     * Creates a normal distribution.
     *
     * @param rng Random number generator.
     * @param mean Mean for this distribution.
     * @param sd Standard deviation for this distribution.
     * @param inverseCumAccuracy Inverse cumulative probability accuracy.
     * @throws NotStrictlyPositiveException if {@code sd <= 0}.
     * @since 3.1
     */
    public NormalDistribution(RandomGenerator rng,
                              double mean,
                              double sd,
                              double inverseCumAccuracy)
        throws NotStrictlyPositiveException {
        super(rng);

        if (sd <= 0) {
            throw new NotStrictlyPositiveException(LocalizedFormats.STANDARD_DEVIATION, sd);
        }

        this.mean = mean;
        standardDeviation = sd;
        logStandardDeviationPlusHalfLog2Pi = FastMath.log(sd) + 0.5 * FastMath.log(2 * FastMath.PI);
        solverAbsoluteAccuracy = inverseCumAccuracy;
    }

    
Access the mean.
Returns:
the mean for this distribution./**
     * Access the mean.
     *
     * @return the mean for this distribution.
     */
    public double getMean() {
        return mean;
    }

    
Access the standard deviation.
Returns:
the standard deviation for this distribution./**
     * Access the standard deviation.
     *
     * @return the standard deviation for this distribution.
     */
    public double getStandardDeviation() {
        return standardDeviation;
    }

    
{@inheritDoc} /** {@inheritDoc} */
    public double density(double x) {
        return FastMath.exp(logDensity(x));
    }

    
{@inheritDoc} /** {@inheritDoc} */
    @Override
    public double logDensity(double x) {
        final double x0 = x - mean;
        final double x1 = x0 / standardDeviation;
        return -0.5 * x1 * x1 - logStandardDeviationPlusHalfLog2Pi;
    }

    
{@inheritDoc} 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. /**
     * {@inheritDoc}
     *
     * If {@code x} is more than 40 standard deviations from the mean, 0 or 1
     * is returned, as in these cases the actual value is within
     * {@code Double.MIN_VALUE} of 0 or 1.
     */
    public double cumulativeProbability(double x)  {
        final double dev = x - mean;
        if (FastMath.abs(dev) > 40 * standardDeviation) {
            return dev < 0 ? 0.0d : 1.0d;
        }
        return 0.5 * Erf.erfc(-dev / (standardDeviation * SQRT2));
    }

    
{@inheritDoc}
Since:
3.2/** {@inheritDoc}
     * @since 3.2
     */
    @Override
    public double inverseCumulativeProbability(final double p) throws OutOfRangeException {
        if (p < 0.0 || p > 1.0) {
            throw new OutOfRangeException(p, 0, 1);
        }
        return mean + standardDeviation * SQRT2 * Erf.erfInv(2 * p - 1);
    }

    
{@inheritDoc}
Deprecated:
See RealDistribution.cumulativeProbability(double, double)/**
     * {@inheritDoc}
     *
     * @deprecated See {@link RealDistribution#cumulativeProbability(double,double)}
     */
    @Override@Deprecated
    public double cumulativeProbability(double x0, double x1)
        throws NumberIsTooLargeException {
        return probability(x0, x1);
    }

    
{@inheritDoc} /** {@inheritDoc} */
    @Override
    public double probability(double x0,
                              double x1)
        throws NumberIsTooLargeException {
        if (x0 > x1) {
            throw new NumberIsTooLargeException(LocalizedFormats.LOWER_ENDPOINT_ABOVE_UPPER_ENDPOINT,
                                                x0, x1, true);
        }
        final double denom = standardDeviation * SQRT2;
        final double v0 = (x0 - mean) / denom;
        final double v1 = (x1 - mean) / denom;
        return 0.5 * Erf.erf(v0, v1);
    }

    
{@inheritDoc} /** {@inheritDoc} */
    @Override
    protected double getSolverAbsoluteAccuracy() {
        return solverAbsoluteAccuracy;
    }

    
{@inheritDoc} For mean parameter mu, the mean is mu. /**
     * {@inheritDoc}
     *
     * For mean parameter {@code mu}, the mean is {@code mu}.
     */
    public double getNumericalMean() {
        return getMean();
    }

    
{@inheritDoc} For standard deviation parameter s, the variance is s^2. /**
     * {@inheritDoc}
     *
     * For standard deviation parameter {@code s}, the variance is {@code s^2}.
     */
    public double getNumericalVariance() {
        final double s = getStandardDeviation();
        return s * s;
    }

    
{@inheritDoc}
The lower bound of the support is always negative infinity
no matter the parameters.
Returns:
lower bound of the support (always Double.NEGATIVE_INFINITY)/**
     * {@inheritDoc}
     *
     * The lower bound of the support is always negative infinity
     * no matter the parameters.
     *
     * @return lower bound of the support (always
     * {@code Double.NEGATIVE_INFINITY})
     */
    public double getSupportLowerBound() {
        return Double.NEGATIVE_INFINITY;
    }

    
{@inheritDoc}
The upper bound of the support is always positive infinity
no matter the parameters.
Returns:
upper bound of the support (always Double.POSITIVE_INFINITY)/**
     * {@inheritDoc}
     *
     * The upper bound of the support is always positive infinity
     * no matter the parameters.
     *
     * @return upper bound of the support (always
     * {@code Double.POSITIVE_INFINITY})
     */
    public double getSupportUpperBound() {
        return Double.POSITIVE_INFINITY;
    }

    
{@inheritDoc} /** {@inheritDoc} */
    public boolean isSupportLowerBoundInclusive() {
        return false;
    }

    
{@inheritDoc} /** {@inheritDoc} */
    public boolean isSupportUpperBoundInclusive() {
        return false;
    }

    
{@inheritDoc}
The support of this distribution is connected.
Returns:
true/**
     * {@inheritDoc}
     *
     * The support of this distribution is connected.
     *
     * @return {@code true}
     */
    public boolean isSupportConnected() {
        return true;
    }

    
{@inheritDoc} /** {@inheritDoc} */
    @Override
    public double sample()  {
        return standardDeviation * random.nextGaussian() + mean;
    }
}