Probability and Random Processes / Edition 1

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An application-based introduction to the statistical analysis of spatially referenced health data Sparked by the growing interest in statistical methods for the analysis of spatially referenced data in the field of public health, Applied Spatial Statistics for Public Health Data fills the need for an introductory, application-oriented text on this timely subject. Written for practicing public health researchers as well as graduate students in related fields, the text provides a thorough introduction to basic concepts and methods in applied spatial statistics as well as a detailed treatment of some of the more recent methods in spatial statistics useful for public health studies that have not been previously covered elsewhere. Assuming minimal knowledge of spatial statistics, the authors provide important statistical approaches for assessing such questions as: Are newly occurring cases of a disease "clustered" in space? Do the cases cluster around suspected sources of increased risk, such as toxic waste sites or other environmental hazards? How do we take monitored pollution concentrations measured at specific locations and interpolate them to locations where no measurements were taken? How do we quantify associations between local disease rates and local exposures? After reviewing traditional statistical methods used in public health research, the text provides an overview of the basic features of spatial data, illustrates various geographic mapping and visualization tools, and describes the sources of publicly available spatial data that might be useful in public health applications.

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Editorial Reviews

From the Publisher
"…an excellent read for students who wish to deepen and enrich their understanding of random processes…" (Technometrics, August 2007)

"A very readable and pleasant book for students and researchers alike." (CHOICE, January 2007)

"This book is of particular value to students and professionals in electrical, computer, and civil engineering, physics, communications, biostatic[s]…and applied mathematics." (PTC Express, March 2007)

"…an exceptional reference…" (Computing, February 1, 2007)

"This book is recommended to libraries and all who are interested in statistics, especially engineers and econometrics." (Chemistry World, August 2007)

"…a single resource to address most issues in probability and random processes." (Zentralblatt MATH, 1105, 85)

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Product Details

Meet the Author

Venkatarama Krishnan, PhD., is Professor Emeritus in the Department of Electrical and Computer Engineering at the University of Massachusetts at Lowell. He has served as a consultant to the Dynamics Research Corporation, the U.S. Department of Transportation, and Bell Laboratories. Dr. Krishnan’s research includes estimation of steady-state queue distribution, tomographic imaging, aerospace, control, communications, and stochastic systems. Dr. Krishnan is a senior member of the IEEE and listed in Who is Who in America.

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Table of Contents


Chapter 1: Sets, Fields, and Events.

1.1 Set Definitions.

1.2 Set Operations.

1.3 Set Algebras, Fields, and Events.

Chapter 2: Probability Space and Axioms.

2.1 Probability Space.

2.2 Conditional Probability.

2.3 Independence.

2.4 Total Probability and Bayes' Theorem.

Chapter 3: Basic Combinatorics.

3.1 Basic Counting Principles.

3.2 Permutations.

3.3 Combinations.

Chapter 4: Discrete Distributions.

4.1 Bernoulli Trials.

4.2 Binomial Distribution.

4.3 Multinomial Distribution.

4.4 Geometric Distribution.

4.5 Negative Binomial Distribution.

4.6 Hypergeometric Distribution.

4.7 Poisson Distribution.

4.8 Logarithmic Distribution.

4.9 Summary of Discrete Distributions.

Chapter 5: Random Variables.

5.1 Definition of Random Variables.

5.2 Determination of Distribution and Density Functions.

5.3 Properties of Distribution and Density Functions.

5.4 Distribution Functions from Density Functions.

Chapter 6: Continuous Random Variables and Basic Distributions.

6.1 Introduction.

6.2 Uniform Distribution.

6.3 Exponential Distribution.

6.4 Normal or Gaussian Distribution.

Chapter 7: Other Continuous Distributions.

7.1 Introduction.

7.2 Triangular Distribution.

7.3 Laplace Distribution.

7.4 Erlang Distribution.

7.5 Gamma Distribution.

7.6 Weibull Distribution.

7.7 Chi-Square Distribution.

7.8 Chi and Other Allied Distributions.

7.9 Student-t Density.

7.10 Snedecor F Distribution.

7.11 Lognormal Distribution.

7.12 Beta Distribution.

7.13 Cauchy Distribution.

7.14 Pareto Distribution.

7.15 Gibbs Distribution.

7.16 Mixed Distributions.

7.17 Summary of Distributions of Continuous Random Variables.

Chapter 8: Conditional Densities and Distributions.

8.1 Conditional Distribution and Density for P(A) = 0.

8.2 Conditional Distribution and Density for P(A) ≠ 0.

8.3 Total Probability and Bayes' Theorem for Densities.

Chapter 9: Joint Densities and Distributions.

9.1 Joint Discrete Distribution Functions.

9.2 Joint Continuous Distribution Functions

9.3 Bivariate Gaussian Distributions.

Chapter 10: Moments and Conditional Moments.

10.1 Expectations.

10.2 Variance.

10.3 Means and Variances of Some Distributions.

10.4 Higher-Order Moments.

10.5 Bivariate Gaussian.

Chapter 11: Characteristic Functions and Generating Functions.

11.1 Characteristic Functions.

11.2 Examples of Characteristic Functions.

11.3 Generating Functions.

11.4 Examples of Generating Functions.

11.5 Moment Generating Functions.

11.6 Cumulant Generating Functions.

11.7 Table of Means and Variances.

Chapter 12: Functions of a Single Random Variable.

12.1 Random Variable g(X).

12.2 Distribution of Y = g(X ).

12.3 Direct Determination of Density fY (y) from fX(x).

12.4 Inverse Problem: Finding g(x) Given fX(x) and fY (y).

12.5 Moments of a Function of a Random Variable.

Chapter 13: Functions of Multiple Random Variables.

13.1 Function of Two Random Variables, Z = g(X,Y ).

13.2 Two Functions of Two Random Variables, Z = g(X,Y ), W = h(X,Y ).

13.3 Direct Determination of Joint Density fZW(z,w ) from fXY(x,y).

13.4 Solving Z = g(X,Y ) Using an Auxiliary Random Variable.

13.5 Multiple Functions of Random Variables.

Chapter 14: Inequalities, Convergences, and Limit Theorems.

14.1 Degenerate Random Variables.

14.2 Chebyshev and Allied Inequalities.

14.3 Markov Inequality.

14.4 Chernoff Bound.

14.5 Cauchy–Schwartz Inequality.

14.6 Jensen’s Inequality.

14.7 Convergence Concepts.

14.8 Limit Theorems.

Chapter 15: Computer Methods for Generating Random Variates.

15.1 Uniform-Distribution Random Variates.

15.2 Histograms.

15.3 Inverse Transformation Techniques.

15.4 Convolution Techniques.

15.5 Acceptance–Rejection Techniques.

Chapter 16: Elements of Matrix Algebra.

16.1 Basic Theory of Matrices.

16.2 Eigenvalues and Eigenvectors of Matrices.

16.3 Vectors and Matrix Differentiations.

16.4 Block Matrices.

Chapter 17: Random Vectors and Mean-Square Estimation.

17.1 Distributions and Densities.

17.2 Moments of Random Vectors.

17.3 Vector Gaussian Random Variables.

17.4 Diagonalization of Covariance Matrices.

17.5 Simultaneous Diagonalization of Covariance Matrices.

17.6 Linear Estimation of Vector Variables.

Chapter 18: Estimation Theory.

18.1 Criteria of Estimators.

18.2 Estimation of Random Variables.

18.3 Estimation of Parameters (Point Estimation).

18.4 Interval Estimation (Confidence Intervals).

18.5 Hypothesis Testing (Binary).

18.6 Bayesian Estimation.

Chapter 19: Random Processes.

19.1 Basic Definitions.

19.2 Stationary Random Processes.

19.3 Ergodic Processes.

19.4 Estimation of Parameters of Random Processes.

19.5 Power Spectral Density.

Chapter 20: Classification of Random Processes.

20.1 Specifications of Random Processes.

20.2 Poisson Process.

20.3 Binomial Process.

20.4 Independent Increment Process.

20.5 Random-Walk Process.

20.6 Gaussian Process.

20.7 Wiener Process (Brownian Motion).

20.8 Markov Process.

20.9 Markov Chain.

20.10 Martingale Process.

20.11 Periodic Random Process.

20.12 Aperiodic Random Process (Karhunen–Loeve Expansion).

Chapter 21: Random Processes and Linear Systems.

21.1 Review of Linear Systems.

21.2 Random Processes through Linear Systems.

21.3 Linear Filters.

21.4 Bandpass Stationary Random Processes.

Chapter 22: Weiner and Kalman Filters.

22.1 Review of Orthogonality Principle.

22.2 Wiener Filtering.

22.3 Discrete Kalman Filter.

22.4 Continuous Kalman Filter.

Chapter 23: Probabilistic Methods in Transmission Tomography.

23.1 Introduction.

23.2 Stochastic Model.

23.3 Stochastic Estimation Algorithm.

23.4 Prior Distribution P(M).

23.5 Computer Simulation.

23.6 Results and Conclusions.

23.7 Discussion of Results.

23.8 References for Chapter 23.


A: A Fourier Transform Tables.

B: Cumulative Gaussian Tables.

C: Inverse Cumulative Gaussian Tables.

D: Inverse Chi-Square Tables.

E: Inverse Student-t Tables.

F: Cumulative Poisson Distribution.

G: Cumulative Binomial Distribution.



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