## Probability and Random ProcessesA resource for probability AND random processes, with hundreds ofworked examples and probability and Fourier transform tables This survival guide in probability and random processes eliminatesthe need to pore through several resources to find a certainformula or table. It offers a compendium of most distributionfunctions used by communication engineers, queuing theoryspecialists, signal processing engineers, biomedical engineers,physicists, and students. Key topics covered include: * Random variables and most of their frequently used discrete andcontinuous probability distribution functions * Moments, transformations, and convergences of randomvariables * Characteristic, generating, and moment-generating functions * Computer generation of random variates * Estimation theory and the associated orthogonalityprinciple * Linear vector spaces and matrix theory with vector and matrixdifferentiation concepts * Vector random variables * Random processes and stationarity concepts * Extensive classification of random processes * Random processes through linear systems and the associated Wienerand Kalman filters * Application of probability in single photon emission tomography(SPECT) More than 400 figures drawn to scale assist readers inunderstanding and applying theory. Many of these figures accompanythe more than 300 examples given to help readers visualize how tosolve the problem at hand. In many instances, worked examples aresolved with more than one approach to illustrate how differentprobability methodologies can work for the same problem. Several probability tables with accuracy up to nine decimal placesare provided in the appendices for quick reference. A specialfeature is the graphical presentation of the commonly occurringFourier transforms, where both time and frequency functions aredrawn to scale. This book is of particular value to undergraduate and graduatestudents in electrical, computer, and civil engineering, as well asstudents in physics and applied mathematics. Engineers, computerscientists, biostatisticians, and researchers in communicationswill also benefit from having a single resource to address mostissues in probability and random processes. |

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### Contents

1 | |

10 | |

25 | |

37 | |

64 | |

CHAPTER 6 Continuous Random Variables and Basic Distributions | 79 |

CHAPTER 7 Other Continuous Distributions | 95 |

CHAPTER 8 Conditional Densities and Distributions | 122 |

CHAPTER 15 Computer Methods for Generating Random Variates | 264 |

CHAPTER 16 Elements of Matrix Algebra | 284 |

CHAPTER 17 Random Vectors and MeanSquare Estimation | 311 |

CHAPTER 18 Estimation Theory | 340 |

CHAPTER 19 Random Processes | 406 |

CHAPTER 20 Classification of Random Processes | 490 |

CHAPTER 21 Random Processes and Linear Systems | 574 |

CHAPTER 22 Weiner and Kalman Filters | 625 |

CHAPTER 9 Joint Densities and Distributions | 135 |

CHAPTER 10 Moments and Conditional Moments | 146 |

CHAPTER 11 Characteristic Functions and Generating Functions | 155 |

CHAPTER 12 Functions of a Single Random Variable | 173 |

CHAPTER 13 Functions of Multiple Random Variables | 206 |

CHAPTER 14 Inequalities Convergences and Limit Theorems | 241 |

CHAPTER 23 Probabilistic Methods in Transmission Tomography | 666 |

APPENDIXES | 683 |

References | 714 |

Index | 716 |

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### Common terms and phrases

ผ ๐ 2006 John Wiley มมม algorithm autocorrelation function autocovariance balls binomial distribution coefﬁcients conditional density conditional expectation conditional probability corresponding covariance CX(h CX(T deﬁned deﬁnition degrees of freedom determine Differentiating digit discrete distribution function eigenvalues equation event Example F X(x ffiffiffi ffiffiffiffi ffiffiffiffiffiffi 2p ffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffi FIGURE ﬁlter ﬁnd ﬁrst follows fX(x fXY(x,y fY(y fZ(z fZW(z,w Gaussian distribution given by Eq Hence histogram impulse response input integral interval inverse Kalman linear Markov chain martingale mean and variance mean value minimum mean-square error noise obtained from Eq orthogonal output parameter pixel Poisson Poisson distribution Probability and Random random process random process X(t region regression result rX(h RX(t RXY(t sample space satisﬁed sequence shown in Fig solution solve stationary process stationary random process Substituting Eq Table uniformly distributed vector Venkatarama Krishnan Copyright Wiener process zero mean

### Popular passages

Page 543 - A system is non-decomposable if every state can be reached from every other state in a finite number of steps, that is, there exists an integer n > 1 such that Pij(n) > 0.

Page 287 - Similarly, the row rank of a matrix is the number of linearly independent rows. If...

Page 154 - Using (2.12) and (2.13), the result that the absolute value of an integral is less than or equal to the integral of the...

Page 293 - The n roots of the characteristic equation are called the eigenvalues of the matrix A, and in linear systems they are also known as the "poles

Page 424 - V , it is necessary and sufficient that the following two conditions be satisfied: 1) EJ ^ 0 does not belong to the code V 2) E.

Page 140 - The joint probability density function of two random variables x and y...

Page 258 - ... 5.5 Let {Xn} be a sequence of independent identically distributed random variables with means 0 and variances 1. Let Sn = Xi + Xz + • • • + Xn be the sequence of their consecutive sums. For n= 1, 2, • • • define a stochastic process {Yn(t), 0 < t < 1} as follows...

Page 283 - The first subscript denotes the row, and the second subscript denotes the column.

Page 5 - Finally, the difference of two sets A and B, denoted by A — B, is the set of all elements of A that are not elements of B. A — B = {x: xe A and x $ B} For example, {1,3, 9} -{3, 5, 7} = {1,9}.