From c71ba35a5358f5ab30307d1c5dfb166cb8f8d97d Mon Sep 17 00:00:00 2001
From: Lakshmi Krishnamurthy <lakshmi7977@gmail.com>
Date: Sat, 12 Aug 2023 00:01:29 -0400
Subject: [PATCH] Features:

	- Composite R^1 Exponential Distribution Shell (1, 2)
	- Minimum Among Exponential Distribution #1 (3, 4)
	- Minimum Among Exponential Distribution #2 (5, 6, 7)
	- Composite R^1 Exponential Distribution Constructor (8, 9, 10)
	- Minimum of Realized Exponential Distributions (11, 12)
	- Composite R^1 Rate Distribution Shell (15, 16)
	- Realized R^1 Rate Distribution Standard (17, 18, 19)
	- Realized R^1 Rate Distribution Constructor (20, 21, 22)
	- R^1 Exponential Rate Distribution Array (23, 24, 25)
	- Probability of Index as Minimum (26, 27, 28)
	- Composite R^1 Order Statistics Shell (29, 30)
	- Exponential IID Joint Moment #1 (31, 32, 33)
	- Exponential IID Joint Moment #2 (34, 35, 36)
	- Exponential IID Joint Moment #3 (37, 38, 39)
	- Exponential IID Joint Moment #4 (40, 41, 42)
	- Two IID Exponential Sum Shell (43, 44)
	- Smaller R^1 Exponential Rate Distribution (45, 46)
	- Larger R^1 Exponential Rate Distribution (47, 48)
	- Two IID Exponential Sum Constructor (49, 50, 51)
	- Two IID Exponential Sum Support (52)
	- Two IID Exponential Sum Density (53, 54, 55)
	- Two IID Sum Quadrature Count (56, 57)
	- Two IID Exponential Sum Mean (58, 59, 60)


Bug Fixes/Re-organization:

	- Composite R^1 To Minima Realization (13, 14)


Samples:


IdeaDRIP:
---
 ReleaseNotes/12_22_2022.txt                   |  37 +++
 ScheduleSheet.xlsx                            | Bin 25702 -> 38267 bytes
 .../CompositeR1OrderStatistics.java           | 175 ++++++++++
 .../exponential/R1RateDistribution.java       |   8 +-
 .../RealizedMinimaR1RateDistribution.java     | 215 ++++++++++++
 .../drip/measure/exponential/TwoIIDSum.java   | 308 ++++++++++++++++++
 6 files changed, 739 insertions(+), 4 deletions(-)
 create mode 100644 ReleaseNotes/12_22_2022.txt
 create mode 100644 src/main/java/org/drip/measure/exponential/CompositeR1OrderStatistics.java
 create mode 100644 src/main/java/org/drip/measure/exponential/RealizedMinimaR1RateDistribution.java
 create mode 100644 src/main/java/org/drip/measure/exponential/TwoIIDSum.java

diff --git a/ReleaseNotes/12_22_2022.txt b/ReleaseNotes/12_22_2022.txt
new file mode 100644
index 000000000000..ea3f40243c84
--- /dev/null
+++ b/ReleaseNotes/12_22_2022.txt
@@ -0,0 +1,37 @@
+
+Features:
+
+	- Composite R^1 Exponential Distribution Shell (1, 2)
+	- Minimum Among Exponential Distribution #1 (3, 4)
+	- Minimum Among Exponential Distribution #2 (5, 6, 7)
+	- Composite R^1 Exponential Distribution Constructor (8, 9, 10)
+	- Minimum of Realized Exponential Distributions (11, 12)
+	- Composite R^1 Rate Distribution Shell (15, 16)
+	- Realized R^1 Rate Distribution Standard (17, 18, 19)
+	- Realized R^1 Rate Distribution Constructor (20, 21, 22)
+	- R^1 Exponential Rate Distribution Array (23, 24, 25)
+	- Probability of Index as Minimum (26, 27, 28)
+	- Composite R^1 Order Statistics Shell (29, 30)
+	- Exponential IID Joint Moment #1 (31, 32, 33)
+	- Exponential IID Joint Moment #2 (34, 35, 36)
+	- Exponential IID Joint Moment #3 (37, 38, 39)
+	- Exponential IID Joint Moment #4 (40, 41, 42)
+	- Two IID Exponential Sum Shell (43, 44)
+	- Smaller R^1 Exponential Rate Distribution (45, 46)
+	- Larger R^1 Exponential Rate Distribution (47, 48)
+	- Two IID Exponential Sum Constructor (49, 50, 51)
+	- Two IID Exponential Sum Support (52)
+	- Two IID Exponential Sum Density (53, 54, 55)
+	- Two IID Sum Quadrature Count (56, 57)
+	- Two IID Exponential Sum Mean (58, 59, 60)
+
+
+Bug Fixes/Re-organization:
+
+	- Composite R^1 To Minima Realization (13, 14)
+
+
+Samples:
+
+
+IdeaDRIP:
diff --git a/ScheduleSheet.xlsx b/ScheduleSheet.xlsx
index 24338f8a20f2037b819edd2b9359b9677767ea98..a535ffd72d2caa087847959ab68c8671a94e1dfa 100644
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diff --git a/src/main/java/org/drip/measure/exponential/CompositeR1OrderStatistics.java b/src/main/java/org/drip/measure/exponential/CompositeR1OrderStatistics.java
new file mode 100644
index 000000000000..1148e406f037
--- /dev/null
+++ b/src/main/java/org/drip/measure/exponential/CompositeR1OrderStatistics.java
@@ -0,0 +1,175 @@
+
+package org.drip.measure.exponential;
+
+/*
+ * -*- mode: java; tab-width: 4; indent-tabs-mode: nil; c-basic-offset: 4 -*-
+ */
+
+/*!
+ * Copyright (C) 2023 Lakshmi Krishnamurthy
+ * 
+ *  This file is part of DROP, an open-source library targeting analytics/risk, transaction cost analytics,
+ *  	asset liability management analytics, capital, exposure, and margin analytics, valuation adjustment
+ *  	analytics, and portfolio construction analytics within and across fixed income, credit, commodity,
+ *  	equity, FX, and structured products. It also includes auxiliary libraries for algorithm support,
+ *  	numerical analysis, numerical optimization, spline builder, model validation, statistical learning,
+ *  	graph builder/navigator, and computational support.
+ *  
+ *  	https://lakshmidrip.github.io/DROP/
+ *  
+ *  DROP is composed of three modules:
+ *  
+ *  - DROP Product Core - https://lakshmidrip.github.io/DROP-Product-Core/
+ *  - DROP Portfolio Core - https://lakshmidrip.github.io/DROP-Portfolio-Core/
+ *  - DROP Computational Core - https://lakshmidrip.github.io/DROP-Computational-Core/
+ * 
+ * 	DROP Product Core implements libraries for the following:
+ * 	- Fixed Income Analytics
+ * 	- Loan Analytics
+ * 	- Transaction Cost Analytics
+ * 
+ * 	DROP Portfolio Core implements libraries for the following:
+ * 	- Asset Allocation Analytics
+ *  - Asset Liability Management Analytics
+ * 	- Capital Estimation Analytics
+ * 	- Exposure Analytics
+ * 	- Margin Analytics
+ * 	- XVA Analytics
+ * 
+ * 	DROP Computational Core implements libraries for the following:
+ * 	- Algorithm Support
+ * 	- Computation Support
+ * 	- Function Analysis
+ *  - Graph Algorithm
+ *  - Model Validation
+ * 	- Numerical Analysis
+ * 	- Numerical Optimizer
+ * 	- Spline Builder
+ *  - Statistical Learning
+ * 
+ * 	Documentation for DROP is Spread Over:
+ * 
+ * 	- Main                     => https://lakshmidrip.github.io/DROP/
+ * 	- Wiki                     => https://github.com/lakshmiDRIP/DROP/wiki
+ * 	- GitHub                   => https://github.com/lakshmiDRIP/DROP
+ * 	- Repo Layout Taxonomy     => https://github.com/lakshmiDRIP/DROP/blob/master/Taxonomy.md
+ * 	- Javadoc                  => https://lakshmidrip.github.io/DROP/Javadoc/index.html
+ * 	- Technical Specifications => https://github.com/lakshmiDRIP/DROP/tree/master/Docs/Internal
+ * 	- Release Versions         => https://lakshmidrip.github.io/DROP/version.html
+ * 	- Community Credits        => https://lakshmidrip.github.io/DROP/credits.html
+ * 	- Issues Catalog           => https://github.com/lakshmiDRIP/DROP/issues
+ * 
+ *  Licensed 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.
+ */
+
+/**
+ * <i>CompositeR1OrderStatistics</i> generates the Order Statistics for a Composite Set of R<sup>1</sup>
+ *  Exponential Distributions. The References are:
+ * 
+ * <br><br>
+ * 	<ul>
+ * 		<li>
+ * 			Devroye, L. (1986): <i>Non-Uniform Random Variate Generation</i> <b>Springer-Verlag</b> New York
+ * 		</li>
+ * 		<li>
+ * 			Exponential Distribution (2019): Exponential Distribution
+ * 				https://en.wikipedia.org/wiki/Exponential_distribution
+ * 		</li>
+ * 		<li>
+ * 			Norton, M., V. Khokhlov, and S. Uryasev (2019): Calculating CVaR and bPOE for Common Probability
+ * 				Distributions with Application to Portfolio Optimization and Density Estimation <i>Annals of
+ * 				Operations Research</i> <b>299 (1-2)</b> 1281-1315
+ * 		</li>
+ * 		<li>
+ * 			Ross, S. M. (2009): <i>Introduction to Probability and Statistics for Engineers and Scientists
+ * 				4<sup>th</sup> Edition</i> <b>Associated Press</b> New York, NY
+ * 		</li>
+ * 		<li>
+ * 			Schmidt, D. F., and D. Makalic (2009): Universal Models for the Exponential Distribution <i>IEEE
+ * 				Transactions on Information Theory</i> <b>55 (7)</b> 3087-3090
+ * 		</li>
+ * 	</ul>
+ *
+ *	<br><br>
+ *  <ul>
+ *		<li><b>Module </b> = <a href = "https://github.com/lakshmiDRIP/DROP/tree/master/ComputationalCore.md">Computational Core Module</a></li>
+ *		<li><b>Library</b> = <a href = "https://github.com/lakshmiDRIP/DROP/tree/master/NumericalAnalysisLibrary.md">Numerical Analysis Library</a></li>
+ *		<li><b>Project</b> = <a href = "https://github.com/lakshmiDRIP/DROP/tree/master/src/main/java/org/drip/measure/README.md">R<sup>d</sup> Continuous/Discrete Probability Measures</a></li>
+ *		<li><b>Package</b> = <a href = "https://github.com/lakshmiDRIP/DROP/tree/master/src/main/java/org/drip/measure/exponential/README.md">R<sup>1</sup> Exponential Distribution Implementation/Properties</a></li>
+ *  </ul>
+ *
+ * @author Lakshmi Krishnamurthy
+ */
+
+public class CompositeR1OrderStatistics
+{
+
+	/**
+	 * Compute the Joint Moment of a Set of i.i.d. Distributions
+	 * 
+	 * @param r1RateDistribution R<sup>1</sup> Exponential Distribution
+	 * @param variateCount Variate Count
+	 * @param index1 First Index
+	 * @param index2 Second Index
+	 * 
+	 * @return Joint Moment of a Set of i.i.d. Distributions
+	 * 
+	 * @throws Exception Thrown if the Inputs are Invalid
+	 */
+
+	public static final double IIDJointMoment (
+		final R1RateDistribution r1RateDistribution,
+		final int variateCount,
+		final int index1,
+		final int index2)
+		throws Exception
+	{
+		if (null == r1RateDistribution ||
+			0 > index1 || index1 >= variateCount ||
+			0 > index2 || index2 >= variateCount || index1 == index2
+		)
+		{
+			throw new Exception (
+				"CompositeR1OrderStatistics::IIDJointMoment => Invalid Inputs"
+			);
+		}
+
+		double expectationMaxIndex = 0.;
+		double expectationMinIndex = 0.;
+		double expectationMinIndexSquared = 0.;
+		int maxIndex = index1 > index2 ? index1 : index2;
+		int minIndex = index1 < index2 ? index1 : index2;
+
+		double inverseRate = 1. / r1RateDistribution.rate();
+
+		for (int k = 0;
+			k < minIndex;
+			++k)
+		{
+			double expectation = inverseRate / (variateCount - k);
+			expectationMinIndexSquared += expectation * expectation;
+			expectationMinIndex += expectation;
+		}
+
+		for (int k = 0;
+			k < maxIndex;
+			++k)
+		{
+			expectationMaxIndex += inverseRate / (variateCount - k);
+		}
+
+		return expectationMinIndex * expectationMaxIndex + expectationMinIndexSquared +
+			expectationMinIndex * expectationMinIndex;
+	}
+}
diff --git a/src/main/java/org/drip/measure/exponential/R1RateDistribution.java b/src/main/java/org/drip/measure/exponential/R1RateDistribution.java
index 180335b1edbb..714bd7d6f8c8 100644
--- a/src/main/java/org/drip/measure/exponential/R1RateDistribution.java
+++ b/src/main/java/org/drip/measure/exponential/R1RateDistribution.java
@@ -163,9 +163,9 @@ public R1RateDistribution (
 	}
 
 	/**
-	 * Retrieve the Rate Parameter Lambda
+	 * Retrieve the Lambda
 	 * 
-	 * @return Rate Parameter Lambda
+	 * @return Lambda
 	 */
 
 	public double lambda()
@@ -174,9 +174,9 @@ public double lambda()
 	}
 
 	/**
-	 * Retrieve the Rate Parameter Lambda
+	 * Retrieve the Rate Parameter
 	 * 
-	 * @return Rate Parameter Lambda
+	 * @return Rate Parameter
 	 */
 
 	public double rate()
diff --git a/src/main/java/org/drip/measure/exponential/RealizedMinimaR1RateDistribution.java b/src/main/java/org/drip/measure/exponential/RealizedMinimaR1RateDistribution.java
new file mode 100644
index 000000000000..eb04e04f7cbe
--- /dev/null
+++ b/src/main/java/org/drip/measure/exponential/RealizedMinimaR1RateDistribution.java
@@ -0,0 +1,215 @@
+
+package org.drip.measure.exponential;
+
+/*
+ * -*- mode: java; tab-width: 4; indent-tabs-mode: nil; c-basic-offset: 4 -*-
+ */
+
+/*!
+ * Copyright (C) 2023 Lakshmi Krishnamurthy
+ * 
+ *  This file is part of DROP, an open-source library targeting analytics/risk, transaction cost analytics,
+ *  	asset liability management analytics, capital, exposure, and margin analytics, valuation adjustment
+ *  	analytics, and portfolio construction analytics within and across fixed income, credit, commodity,
+ *  	equity, FX, and structured products. It also includes auxiliary libraries for algorithm support,
+ *  	numerical analysis, numerical optimization, spline builder, model validation, statistical learning,
+ *  	graph builder/navigator, and computational support.
+ *  
+ *  	https://lakshmidrip.github.io/DROP/
+ *  
+ *  DROP is composed of three modules:
+ *  
+ *  - DROP Product Core - https://lakshmidrip.github.io/DROP-Product-Core/
+ *  - DROP Portfolio Core - https://lakshmidrip.github.io/DROP-Portfolio-Core/
+ *  - DROP Computational Core - https://lakshmidrip.github.io/DROP-Computational-Core/
+ * 
+ * 	DROP Product Core implements libraries for the following:
+ * 	- Fixed Income Analytics
+ * 	- Loan Analytics
+ * 	- Transaction Cost Analytics
+ * 
+ * 	DROP Portfolio Core implements libraries for the following:
+ * 	- Asset Allocation Analytics
+ *  - Asset Liability Management Analytics
+ * 	- Capital Estimation Analytics
+ * 	- Exposure Analytics
+ * 	- Margin Analytics
+ * 	- XVA Analytics
+ * 
+ * 	DROP Computational Core implements libraries for the following:
+ * 	- Algorithm Support
+ * 	- Computation Support
+ * 	- Function Analysis
+ *  - Graph Algorithm
+ *  - Model Validation
+ * 	- Numerical Analysis
+ * 	- Numerical Optimizer
+ * 	- Spline Builder
+ *  - Statistical Learning
+ * 
+ * 	Documentation for DROP is Spread Over:
+ * 
+ * 	- Main                     => https://lakshmidrip.github.io/DROP/
+ * 	- Wiki                     => https://github.com/lakshmiDRIP/DROP/wiki
+ * 	- GitHub                   => https://github.com/lakshmiDRIP/DROP
+ * 	- Repo Layout Taxonomy     => https://github.com/lakshmiDRIP/DROP/blob/master/Taxonomy.md
+ * 	- Javadoc                  => https://lakshmidrip.github.io/DROP/Javadoc/index.html
+ * 	- Technical Specifications => https://github.com/lakshmiDRIP/DROP/tree/master/Docs/Internal
+ * 	- Release Versions         => https://lakshmidrip.github.io/DROP/version.html
+ * 	- Community Credits        => https://lakshmidrip.github.io/DROP/credits.html
+ * 	- Issues Catalog           => https://github.com/lakshmiDRIP/DROP/issues
+ * 
+ *  Licensed 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.
+ */
+
+/**
+ * <i>RealizedMinimaR1RateDistribution</i> implements the Rate Parameterization of the Realized Minimum among
+ *  the Set of R<sup>1</sup> Exponential Distributions. The References are:
+ * 
+ * <br><br>
+ * 	<ul>
+ * 		<li>
+ * 			Devroye, L. (1986): <i>Non-Uniform Random Variate Generation</i> <b>Springer-Verlag</b> New York
+ * 		</li>
+ * 		<li>
+ * 			Exponential Distribution (2019): Exponential Distribution
+ * 				https://en.wikipedia.org/wiki/Exponential_distribution
+ * 		</li>
+ * 		<li>
+ * 			Norton, M., V. Khokhlov, and S. Uryasev (2019): Calculating CVaR and bPOE for Common Probability
+ * 				Distributions with Application to Portfolio Optimization and Density Estimation <i>Annals of
+ * 				Operations Research</i> <b>299 (1-2)</b> 1281-1315
+ * 		</li>
+ * 		<li>
+ * 			Ross, S. M. (2009): <i>Introduction to Probability and Statistics for Engineers and Scientists
+ * 				4<sup>th</sup> Edition</i> <b>Associated Press</b> New York, NY
+ * 		</li>
+ * 		<li>
+ * 			Schmidt, D. F., and D. Makalic (2009): Universal Models for the Exponential Distribution <i>IEEE
+ * 				Transactions on Information Theory</i> <b>55 (7)</b> 3087-3090
+ * 		</li>
+ * 	</ul>
+ *
+ *	<br><br>
+ *  <ul>
+ *		<li><b>Module </b> = <a href = "https://github.com/lakshmiDRIP/DROP/tree/master/ComputationalCore.md">Computational Core Module</a></li>
+ *		<li><b>Library</b> = <a href = "https://github.com/lakshmiDRIP/DROP/tree/master/NumericalAnalysisLibrary.md">Numerical Analysis Library</a></li>
+ *		<li><b>Project</b> = <a href = "https://github.com/lakshmiDRIP/DROP/tree/master/src/main/java/org/drip/measure/README.md">R<sup>d</sup> Continuous/Discrete Probability Measures</a></li>
+ *		<li><b>Package</b> = <a href = "https://github.com/lakshmiDRIP/DROP/tree/master/src/main/java/org/drip/measure/exponential/README.md">R<sup>1</sup> Exponential Distribution Implementation/Properties</a></li>
+ *  </ul>
+ *
+ * @author Lakshmi Krishnamurthy
+ */
+
+public class RealizedMinimaR1RateDistribution
+	extends R1RateDistribution
+{
+	private R1RateDistribution[] _r1RateDistributionArray = null;
+
+	/**
+	 * Standard Instance of RealizedMinimaR1RateDistribution
+	 * 
+	 * @param r1RateDistributionArray R<sup>1</sup> Exponential Distribution Array
+	 * 
+	 * @return Standard Instance of RealizedMinimaR1RateDistribution
+	 */
+
+	public static final RealizedMinimaR1RateDistribution Standard (
+		final R1RateDistribution[] r1RateDistributionArray)
+		throws Exception
+	{
+		if (null == r1RateDistributionArray)
+		{
+			throw new Exception (
+				"RealizedMinimaR1RateDistribution Constructor => Invalid Input"
+			);
+		}
+
+		double compositeDistributionLambda = 0.;
+		int exponentialDistributionCount = r1RateDistributionArray.length;
+
+		if (0 == exponentialDistributionCount)
+		{
+			return null;
+		}
+
+		try
+		{
+			for (int distributionIndex = 0;
+				distributionIndex < exponentialDistributionCount;
+				++distributionIndex)
+			{
+				compositeDistributionLambda += r1RateDistributionArray[distributionIndex].rate();
+			}
+
+			return new RealizedMinimaR1RateDistribution (
+				r1RateDistributionArray,
+				compositeDistributionLambda
+			);
+		}
+		catch (Exception e)
+		{
+			e.printStackTrace();
+		}
+
+		return null;
+	}
+
+	private RealizedMinimaR1RateDistribution (
+		final R1RateDistribution[] r1RateDistributionArray,
+		final double compositeDistributionLambda)
+		throws Exception
+	{
+		super (
+			compositeDistributionLambda
+		);
+
+		_r1RateDistributionArray = r1RateDistributionArray;
+	}
+
+	/**
+	 * Retrieve the R<sup>1</sup> Exponential Distribution Array
+	 * 
+	 * @return The R<sup>1</sup> Exponential Distribution Array
+	 */
+
+	public R1RateDistribution[] r1RateDistributionArray()
+	{
+		return _r1RateDistributionArray;
+	}
+
+	/**
+	 * Calculate the Probability that the specified Index corresponds to the Realized Minimum
+	 * 
+	 * @param index Specified Index
+	 * 
+	 * @return Probability that the specified Index corresponds to the Realized Minimum
+	 * 
+	 * @throws Exception Thrown if the Index cannot be calculated
+	 */
+
+	public double probabilityOfIndexAsMinimum (
+		final int index)
+		throws Exception
+	{
+		if (index >= _r1RateDistributionArray.length)
+		{
+			throw new Exception (
+				"RealizedMinimaR1RateDistribution::probabilityOfIndexAsMinimum => Invalid Index"
+			);
+		}
+
+		return _r1RateDistributionArray[index].rate() / rate();
+	}
+}
diff --git a/src/main/java/org/drip/measure/exponential/TwoIIDSum.java b/src/main/java/org/drip/measure/exponential/TwoIIDSum.java
new file mode 100644
index 000000000000..2e017e6b1735
--- /dev/null
+++ b/src/main/java/org/drip/measure/exponential/TwoIIDSum.java
@@ -0,0 +1,308 @@
+
+package org.drip.measure.exponential;
+
+import org.drip.function.definition.R1ToR1;
+import org.drip.measure.continuous.R1Univariate;
+import org.drip.numerical.integration.NewtonCotesQuadratureGenerator;
+import org.drip.numerical.integration.R1ToR1Integrator;
+
+/*
+ * -*- mode: java; tab-width: 4; indent-tabs-mode: nil; c-basic-offset: 4 -*-
+ */
+
+/*!
+ * Copyright (C) 2023 Lakshmi Krishnamurthy
+ * 
+ *  This file is part of DROP, an open-source library targeting analytics/risk, transaction cost analytics,
+ *  	asset liability management analytics, capital, exposure, and margin analytics, valuation adjustment
+ *  	analytics, and portfolio construction analytics within and across fixed income, credit, commodity,
+ *  	equity, FX, and structured products. It also includes auxiliary libraries for algorithm support,
+ *  	numerical analysis, numerical optimization, spline builder, model validation, statistical learning,
+ *  	graph builder/navigator, and computational support.
+ *  
+ *  	https://lakshmidrip.github.io/DROP/
+ *  
+ *  DROP is composed of three modules:
+ *  
+ *  - DROP Product Core - https://lakshmidrip.github.io/DROP-Product-Core/
+ *  - DROP Portfolio Core - https://lakshmidrip.github.io/DROP-Portfolio-Core/
+ *  - DROP Computational Core - https://lakshmidrip.github.io/DROP-Computational-Core/
+ * 
+ * 	DROP Product Core implements libraries for the following:
+ * 	- Fixed Income Analytics
+ * 	- Loan Analytics
+ * 	- Transaction Cost Analytics
+ * 
+ * 	DROP Portfolio Core implements libraries for the following:
+ * 	- Asset Allocation Analytics
+ *  - Asset Liability Management Analytics
+ * 	- Capital Estimation Analytics
+ * 	- Exposure Analytics
+ * 	- Margin Analytics
+ * 	- XVA Analytics
+ * 
+ * 	DROP Computational Core implements libraries for the following:
+ * 	- Algorithm Support
+ * 	- Computation Support
+ * 	- Function Analysis
+ *  - Graph Algorithm
+ *  - Model Validation
+ * 	- Numerical Analysis
+ * 	- Numerical Optimizer
+ * 	- Spline Builder
+ *  - Statistical Learning
+ * 
+ * 	Documentation for DROP is Spread Over:
+ * 
+ * 	- Main                     => https://lakshmidrip.github.io/DROP/
+ * 	- Wiki                     => https://github.com/lakshmiDRIP/DROP/wiki
+ * 	- GitHub                   => https://github.com/lakshmiDRIP/DROP
+ * 	- Repo Layout Taxonomy     => https://github.com/lakshmiDRIP/DROP/blob/master/Taxonomy.md
+ * 	- Javadoc                  => https://lakshmidrip.github.io/DROP/Javadoc/index.html
+ * 	- Technical Specifications => https://github.com/lakshmiDRIP/DROP/tree/master/Docs/Internal
+ * 	- Release Versions         => https://lakshmidrip.github.io/DROP/version.html
+ * 	- Community Credits        => https://lakshmidrip.github.io/DROP/credits.html
+ * 	- Issues Catalog           => https://github.com/lakshmiDRIP/DROP/issues
+ * 
+ *  Licensed 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.
+ */
+
+/**
+ * <i>TwoIIDSum</i> implements the PDF of the Sum of Two IID Exponential Random Variables. The References
+ *  are:
+ * 
+ * <br><br>
+ * 	<ul>
+ * 		<li>
+ * 			Devroye, L. (1986): <i>Non-Uniform Random Variate Generation</i> <b>Springer-Verlag</b> New York
+ * 		</li>
+ * 		<li>
+ * 			Exponential Distribution (2019): Exponential Distribution
+ * 				https://en.wikipedia.org/wiki/Exponential_distribution
+ * 		</li>
+ * 		<li>
+ * 			Norton, M., V. Khokhlov, and S. Uryasev (2019): Calculating CVaR and bPOE for Common Probability
+ * 				Distributions with Application to Portfolio Optimization and Density Estimation <i>Annals of
+ * 				Operations Research</i> <b>299 (1-2)</b> 1281-1315
+ * 		</li>
+ * 		<li>
+ * 			Ross, S. M. (2009): <i>Introduction to Probability and Statistics for Engineers and Scientists
+ * 				4<sup>th</sup> Edition</i> <b>Associated Press</b> New York, NY
+ * 		</li>
+ * 		<li>
+ * 			Schmidt, D. F., and D. Makalic (2009): Universal Models for the Exponential Distribution <i>IEEE
+ * 				Transactions on Information Theory</i> <b>55 (7)</b> 3087-3090
+ * 		</li>
+ * 	</ul>
+ *
+ *	<br><br>
+ *  <ul>
+ *		<li><b>Module </b> = <a href = "https://github.com/lakshmiDRIP/DROP/tree/master/ComputationalCore.md">Computational Core Module</a></li>
+ *		<li><b>Library</b> = <a href = "https://github.com/lakshmiDRIP/DROP/tree/master/NumericalAnalysisLibrary.md">Numerical Analysis Library</a></li>
+ *		<li><b>Project</b> = <a href = "https://github.com/lakshmiDRIP/DROP/tree/master/src/main/java/org/drip/measure/README.md">R<sup>d</sup> Continuous/Discrete Probability Measures</a></li>
+ *		<li><b>Package</b> = <a href = "https://github.com/lakshmiDRIP/DROP/tree/master/src/main/java/org/drip/measure/exponential/README.md">R<sup>1</sup> Exponential Distribution Implementation/Properties</a></li>
+ *  </ul>
+ *
+ * @author Lakshmi Krishnamurthy
+ */
+
+public class TwoIIDSum
+	extends R1Univariate
+{
+	private static final int QUADRATURE_POINT_COUNT = 100;
+
+	private R1RateDistribution _largerR1RateDistribution = null;
+	private R1RateDistribution _smallerR1RateDistribution = null;
+
+	/**
+	 * TwoIIDSum Constructor
+	 * 
+	 * @param firstR1RateDistribution First R<sup>1</sup> Exponential Distribution
+	 * @param secondR1RateDistribution Second R<sup>1</sup> Exponential Distribution
+	 * 
+	 * @throws Exception Thrown if Inputs are Invalid
+	 */
+
+	public TwoIIDSum (
+		final R1RateDistribution firstR1RateDistribution,
+		final R1RateDistribution secondR1RateDistribution)
+		throws Exception
+	{
+		if (null == firstR1RateDistribution || null == secondR1RateDistribution)
+		{
+			throw new Exception (
+				"TwoIIDSum Constructor: Invalid Inputs"
+			);
+		}
+
+		double firstRate = firstR1RateDistribution.rate();
+
+		double secondRate = secondR1RateDistribution.rate();
+
+		if (firstRate > secondRate)
+		{
+			_largerR1RateDistribution = secondR1RateDistribution;
+			_smallerR1RateDistribution = firstR1RateDistribution;
+		}
+		else
+		{
+			_largerR1RateDistribution = firstR1RateDistribution;
+			_smallerR1RateDistribution = secondR1RateDistribution;
+		}
+	}
+
+	/**
+	 * Retrieve the Larger Exponential Distribution
+	 * 
+	 * @return The Larger Exponential Distribution
+	 */
+
+	public R1RateDistribution largerR1RateDistribution()
+	{
+		return _largerR1RateDistribution;
+	}
+
+	/**
+	 * Retrieve the Smaller Exponential Distribution
+	 * 
+	 * @return The Smaller Exponential Distribution
+	 */
+
+	public R1RateDistribution smallerR1RateDistribution()
+	{
+		return _smallerR1RateDistribution;
+	}
+
+	@Override public double[] support()
+	{
+		return new double[]
+		{
+			0.,
+			Double.POSITIVE_INFINITY
+		};
+	}
+
+	@Override public double density (
+		final double t)
+		throws Exception
+	{
+		if (!supported (
+			t
+		))
+		{
+			throw new Exception (
+				"TwoIIDSum::density => Variate not in Range"
+			);
+		}
+
+		double largerRate = _largerR1RateDistribution.rate();
+
+		double smallerRate = _smallerR1RateDistribution.rate();
+
+		if (largerRate == smallerRate)
+		{
+			return smallerRate * smallerRate * t * Math.exp (
+				-1. * smallerRate * t
+			);
+		}
+
+		return smallerRate * largerRate * (
+			Math.exp (
+				-1. * smallerRate * t
+			) - Math.exp (
+				-1. * largerRate * t
+			)
+		) / (largerRate - smallerRate);
+	}
+
+	@Override public double mean()
+		throws Exception
+	{
+		return NewtonCotesQuadratureGenerator.GaussLaguerreRightDefinite (
+			0.,
+			QUADRATURE_POINT_COUNT
+		).integrate (
+			new R1ToR1 (
+				null
+			)
+			{
+				@Override public double evaluate (
+					final double z)
+					throws Exception
+				{
+					return z * density (
+						z
+					);
+				}
+			}
+		);
+	}
+
+	@Override public double variance()
+		throws Exception
+	{
+		final double mean = mean();
+
+		return NewtonCotesQuadratureGenerator.GaussLaguerreRightDefinite (
+			0.,
+			QUADRATURE_POINT_COUNT
+		).integrate (
+			new R1ToR1 (
+				null
+			)
+			{
+				@Override public double evaluate (
+					final double z)
+					throws Exception
+				{
+					return (z - mean) * (z - mean) * density (
+						z
+					);
+				}
+			}
+		);
+	}
+
+	@Override public double cumulative (
+		final double upper)
+		throws Exception
+	{
+		if (Double.isNaN (
+			upper
+		))
+		{
+			throw new Exception (
+				"TwoIIDSum::cumulative => Invalid Upper Variate"
+			);
+		}
+
+		return R1ToR1Integrator.Boole (
+			new R1ToR1 (
+				null
+			)
+			{
+				@Override public double evaluate (
+					final double z)
+					throws Exception
+				{
+					return density (
+						z
+					);
+				}
+			},
+			0.,
+			upper
+		);
+	}
+}