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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.
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package org.apache.commons.math3.distribution;

import org.apache.commons.math3.exception.NotStrictlyPositiveException;
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.Beta;
import org.apache.commons.math3.util.CombinatoricsUtils;
import org.apache.commons.math3.util.FastMath;

/**
 * 

* Implementation of the Pascal distribution. The Pascal distribution is a * special case of the Negative Binomial distribution where the number of * successes parameter is an integer. *

*

* There are various ways to express the probability mass and distribution * functions for the Pascal distribution. The present implementation represents * the distribution of the number of failures before {@code r} successes occur. * This is the convention adopted in e.g. * MathWorld, * but not in * Wikipedia. *

*

* For a random variable {@code X} whose values are distributed according to this * distribution, the probability mass function is given by
* {@code P(X = k) = C(k + r - 1, r - 1) * p^r * (1 - p)^k,}
* where {@code r} is the number of successes, {@code p} is the probability of * success, and {@code X} is the total number of failures. {@code C(n, k)} is * the binomial coefficient ({@code n} choose {@code k}). The mean and variance * of {@code X} are
* {@code E(X) = (1 - p) * r / p, var(X) = (1 - p) * r / p^2.}
* Finally, the cumulative distribution function is given by
* {@code P(X <= k) = I(p, r, k + 1)}, * where I is the regularized incomplete Beta function. *

* * @see * Negative binomial distribution (Wikipedia) * @see * Negative binomial distribution (MathWorld) * @since 1.2 (changed to concrete class in 3.0) */ public class PascalDistribution extends AbstractIntegerDistribution { /** Serializable version identifier. */ private static final long serialVersionUID = 6751309484392813623L; /** The number of successes. */ private final int numberOfSuccesses; /** The probability of success. */ private final double probabilityOfSuccess; /** The value of {@code log(p)}, where {@code p} is the probability of success, * stored for faster computation. */ private final double logProbabilityOfSuccess; /** The value of {@code log(1-p)}, where {@code p} is the probability of success, * stored for faster computation. */ private final double log1mProbabilityOfSuccess; /** * Create a Pascal distribution with the given number of successes and * probability of success. *

* Note: 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 r Number of successes. * @param p Probability of success. * @throws NotStrictlyPositiveException if the number of successes is not positive * @throws OutOfRangeException if the probability of success is not in the * range {@code [0, 1]}. */ public PascalDistribution(int r, double p) throws NotStrictlyPositiveException, OutOfRangeException { this(new Well19937c(), r, p); } /** * Create a Pascal distribution with the given number of successes and * probability of success. * * @param rng Random number generator. * @param r Number of successes. * @param p Probability of success. * @throws NotStrictlyPositiveException if the number of successes is not positive * @throws OutOfRangeException if the probability of success is not in the * range {@code [0, 1]}. * @since 3.1 */ public PascalDistribution(RandomGenerator rng, int r, double p) throws NotStrictlyPositiveException, OutOfRangeException { super(rng); if (r <= 0) { throw new NotStrictlyPositiveException(LocalizedFormats.NUMBER_OF_SUCCESSES, r); } if (p < 0 || p > 1) { throw new OutOfRangeException(p, 0, 1); } numberOfSuccesses = r; probabilityOfSuccess = p; logProbabilityOfSuccess = FastMath.log(p); log1mProbabilityOfSuccess = FastMath.log1p(-p); } /** * Access the number of successes for this distribution. * * @return the number of successes. */ public int getNumberOfSuccesses() { return numberOfSuccesses; } /** * Access the probability of success for this distribution. * * @return the probability of success. */ public double getProbabilityOfSuccess() { return probabilityOfSuccess; } /** {@inheritDoc} */ public double probability(int x) { double ret; if (x < 0) { ret = 0.0; } else { ret = CombinatoricsUtils.binomialCoefficientDouble(x + numberOfSuccesses - 1, numberOfSuccesses - 1) * FastMath.pow(probabilityOfSuccess, numberOfSuccesses) * FastMath.pow(1.0 - probabilityOfSuccess, x); } return ret; } /** {@inheritDoc} */ @Override public double logProbability(int x) { double ret; if (x < 0) { ret = Double.NEGATIVE_INFINITY; } else { ret = CombinatoricsUtils.binomialCoefficientLog(x + numberOfSuccesses - 1, numberOfSuccesses - 1) + logProbabilityOfSuccess * numberOfSuccesses + log1mProbabilityOfSuccess * x; } return ret; } /** {@inheritDoc} */ public double cumulativeProbability(int x) { double ret; if (x < 0) { ret = 0.0; } else { ret = Beta.regularizedBeta(probabilityOfSuccess, numberOfSuccesses, x + 1.0); } return ret; } /** * {@inheritDoc} * * For number of successes {@code r} and probability of success {@code p}, * the mean is {@code r * (1 - p) / p}. */ public double getNumericalMean() { final double p = getProbabilityOfSuccess(); final double r = getNumberOfSuccesses(); return (r * (1 - p)) / p; } /** * {@inheritDoc} * * For number of successes {@code r} and probability of success {@code p}, * the variance is {@code r * (1 - p) / p^2}. */ public double getNumericalVariance() { final double p = getProbabilityOfSuccess(); final double r = getNumberOfSuccesses(); return r * (1 - p) / (p * p); } /** * {@inheritDoc} * * The lower bound of the support is always 0 no matter the parameters. * * @return lower bound of the support (always 0) */ public int getSupportLowerBound() { return 0; } /** * {@inheritDoc} * * The upper bound of the support is always positive infinity no matter the * parameters. Positive infinity is symbolized by {@code Integer.MAX_VALUE}. * * @return upper bound of the support (always {@code Integer.MAX_VALUE} * for positive infinity) */ public int getSupportUpperBound() { return Integer.MAX_VALUE; } /** * {@inheritDoc} * * The support of this distribution is connected. * * @return {@code true} */ public boolean isSupportConnected() { return true; } }





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