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2.2.3 Continuous Random Variables and Probability Density
Functions 53
2.2.4 Classes of Discrete and Continuous PDFs 57
2.2.5 Mixed Discrete-Continuous Random Variables 60
2.3. Univariate Cumulative Distribution Functions 62
2.3.1 CDF Properties 64
2.3.2 Duality Between CDFs and PDFs 66
2.4. Multivariate Random Variables, PDFs, and CDFs 68
2.4.1 Multivariate Random Variable Properties
and Classes of PDFs 70
2.4.2 Multivariate CDFs and Duality with PDFs 74
2.4.3 Multivariate Mixed Discrete-Continuous
and Composite Random Variables 78
2.5. Marginal Probability Density Functions and CDFs 78
2.5.1 Bivariate Case 79
2.5.2 n-Variate Case 82
2.5.3 Marginal Cumulative Distribution Functions (MCDFs) 84
2.6. Conditional Density Functions 84
2.6.1 Bivariate Case 85
2.6.2 Conditioning on Elementary Events in Continuous
Cases-Bivariate 87
2.6.3 Conditioning on Elementary Events in Continuous
Cases: n-Variate Case 89
2.6.4 Conditional CDFs 90
2.7. Independence of Random Variables 90
2.7.1 Bivariate Case 91
2.7.2 n-Variate 94
2.7.3 Marginal Densities Do Not Determine an n-Variate
Density Without Independence 95
2.7.4 Independence Between Random Vectors and Between
Functions of Random Vectors 96
2.8. Extended Example of Multivariate Concepts
in the Continuous Case 98
Keywords, Phrases, and Symbols 100
Problems 101
3. Expectations and Moments of Random Variables 111
3.1. Expectation of a Random Variable 111
3.2. Stieltjes Integration and Implicit Support Convention 117
3.3. Expectation of a Function of Random Variables 118
3.4. Expectation Properties 122
3.4.1 Scalar Random Variables 122
3.4.2 Multivariate Random Variables
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