Monte Carlo Methods Utilizing Mathematica®: Applications in Inverse Transform and Acceptance-Rejection Sampling

1
	Preface
	Contents
1 (1)
	Part I Monte Carlo Methods Using Inverse Transform Sampling Utilizing Mathematica
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	1 Introduction for Part I
		1.1 Random Variable
		1.2 Continuous Random Variable
		1.3 Uniform Random Variable
		1.4 Normal or Gaussian Random Variable
		1.5 Generating Random Variates by Inverse Transform Method
		1.6 Variance Reduction and Importance Sampling
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	2 Evaluation of Definite Integrals Using Inverse Transform Sampling Utilizing Mathematica
		2.1 Evaluation of Definite Integrals Using Inverse Transform Sampling: Example I
		2.2 Evaluation of Definite Integrals Using Inverse Transform Sampling: Example II
		2.3 Evaluation of Definite Integrals Using Inverse Transform Sampling: Example III
		2.4 Evaluation of Definite Integrals Using Inverse Transform Sampling: Example IV
		2.5 Evaluation of Definite Integrals Using Inverse Transform Sampling: Example V
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	Part II Monte Carlo Methods Using Acceptance-Rejection Sampling Utilizing Mathematica
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	3 Introduction for Part II
		3.1 Random Variable
		3.2 Continuous Random Variable
		3.3 Uniform Random Variable
		3.4 Normal or Gaussian Random Variable
		3.5 Generating Random Variates by Acceptance-Rejection Method
		3.6 Variance Reduction and Importance Sampling
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	4 Evaluation of Definite Integrals Using Acceptance-Rejection Sampling Utilizing Mathematica
		4.1 Evaluation of Definite Integrals Using Acceptance-Rejection Sampling: Example I
		4.2 Evaluation of Definite Integrals Using Acceptance-Rejection Sampling: Example II
		4.3 Evaluation of Definite Integrals Using Acceptance-Rejection Sampling: Example III
		4.4 Evaluation of Definite Integrals Using Acceptance-Rejection Sampling: Example IV
		4.5 Evaluation of Definite Integrals Using Acceptance-Rejection Sampling: Example V
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	5 Variational Monte Carlo Method Applied to Ground State of Simple Harmonic Oscillator Using Acceptance-Rejection Sampling Utilizing Mathematica
		5.1 The Variational Method of Quantum Mechanics Applied to Ground State of Any Quantum Mechanical System
		5.2 The Variational Method of Quantum Mechanics Applied to Ground State of Simple Harmonic Oscillator
		5.3 Ground State Energy of Simple Harmonic Oscillator Using Variational Quantum Monte Carlo Method and Acceptance-Rejection Sampling
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	Concluding Remarks
	Appendix: Handouts for Computational Lab and Automated Collection of Accepted Random Variates
		A.1 Experiment I: Monte Carlo Integration Using Inverse Transform Sampling Using (a) Mathematica and (b) MS Excel
		A.2 Experiment II: Monte Carlo Integration Using Acceptance-Rejection Sampling, Gaussian Pdf and Mathematica
		A.3 Automated Collection of Accepted Random Variates
	References