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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
+ * limitations under the License.
+ */
+
+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 <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
+     */
+    public static final double DEFAULT_INVERSE_ABSOLUTE_ACCURACY = 1e-9;
+
+    /** Serializable version identifier. */
+    private static final long serialVersionUID = 8589540077390120676L;
+
+    /** &radic;(2) */
+    private static final double SQRT2 = FastMath.sqrt(2.0);
+
+    /** Mean of this distribution. */
+    private final double mean;
+
+    /** Standard deviation of this distribution. */
+    private final double standardDeviation;
+
+    /** The value of {@code log(sd) + 0.5*log(2*pi)} stored for faster computation. */
+    private final double logStandardDeviationPlusHalfLog2Pi;
+
+    /** Inverse cumulative probability accuracy. */
+    private final double solverAbsoluteAccuracy;
+
+    /**
+     * 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.
+     *
+     * <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.
+     *
+     * <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.
+     *
+     * @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.
+     *
+     * @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.
+     *
+     * @return the mean for this distribution.
+     */
+    public double getMean() {
+        return mean;
+    }
+
+    /**
+     * Access the standard deviation.
+     *
+     * @return the standard deviation for this distribution.
+     */
+    public double getStandardDeviation() {
+        return standardDeviation;
+    }
+
+    /** {@inheritDoc} */
+    public double density(double x) {
+        return FastMath.exp(logDensity(x));
+    }
+
+    /** {@inheritDoc} */
+    @Override
+    public double logDensity(double x) {
+        final double x0 = x - mean;
+        final double x1 = x0 / standardDeviation;
+        return -0.5 * x1 * x1 - logStandardDeviationPlusHalfLog2Pi;
+    }
+
+    /**
+     * {@inheritDoc}
+     *
+     * <p>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
+     */
+    @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 {@link RealDistribution#cumulativeProbability(double,double)}
+     */
+    @Override
+    @Deprecated
+    public double cumulativeProbability(double x0, double x1) throws NumberIsTooLargeException {
+        return probability(x0, x1);
+    }
+
+    /** {@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} */
+    @Override
+    protected double getSolverAbsoluteAccuracy() {
+        return solverAbsoluteAccuracy;
+    }
+
+    /**
+     * {@inheritDoc}
+     *
+     * <p>For mean parameter {@code mu}, the mean is {@code mu}.
+     */
+    public double getNumericalMean() {
+        return getMean();
+    }
+
+    /**
+     * {@inheritDoc}
+     *
+     * <p>For standard deviation parameter {@code s}, the variance is {@code s^2}.
+     */
+    public double getNumericalVariance() {
+        final double s = getStandardDeviation();
+        return s * s;
+    }
+
+    /**
+     * {@inheritDoc}
+     *
+     * <p>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}
+     *
+     * <p>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} */
+    public boolean isSupportLowerBoundInclusive() {
+        return false;
+    }
+
+    /** {@inheritDoc} */
+    public boolean isSupportUpperBoundInclusive() {
+        return false;
+    }
+
+    /**
+     * {@inheritDoc}
+     *
+     * <p>The support of this distribution is connected.
+     *
+     * @return {@code true}
+     */
+    public boolean isSupportConnected() {
+        return true;
+    }
+
+    /** {@inheritDoc} */
+    @Override
+    public double sample() {
+        return standardDeviation * random.nextGaussian() + mean;
+    }
+}