note01_probability

# note01_probability - Contents 2 Mathematical Preliminaries...

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Unformatted text preview: Contents 2 Mathematical Preliminaries 4 2.1 Some Probability Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.1.1 Conditional Probability . . . . . . . . . . . . . . . . . . . . . . . 8 2.1.2 The Law of Total Probability . . . . . . . . . . . . . . . . . . . . 8 2.1.3 Statistical Independence of Events . . . . . . . . . . . . . . . . . 9 2.1.4 Expectation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.1.5 Random Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.1.6 Independence of Random Variables . . . . . . . . . . . . . . . . . 11 2.1.7 Uncorrelatedness . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.1.8 The Characteristic Function . . . . . . . . . . . . . . . . . . . . . 12 2.2 Stochastic Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.2.1 Autocovariance and Autocorrelation Function . . . . . . . . . . . 14 2.2.2 The Spectral Density at the Output of a Linear Filter . . . . . . 17 1 List of Figures 2.1 The Gaussian, Laplacian and uniform density functions. . . . . . . . . . 8 2.2 A simple model for a binary digital communication system. . . . . . . . 9 2.3 Realizations of stochastic processes. . . . . . . . . . . . . . . . . . . . . 15 2.4 The zero-mean Gaussian density at different variances. . . . . . . . . . . 15 2.5 The spectral density of white Gaussian noise. . . . . . . . . . . . . . . . 17 2.6 The output of a linear filter in response to a stochastic input. . . . . . . 17 2 List of Tables 3 Chapter 2 Mathematical Preliminaries 2.1 Some Probability Theory To introduce basic notation and concepts some simple set- theoretic definitions are given next. Definition 1 (a set) A set is a collection of objects, known as elements. Examples of sets are: A = { , 1 } Z = { ,- 1 , , 1 , 2 , } , the set of all integers. R = { x ;- < x < } , the set of real numbers. R + = { x ;0 < x < } , the set of positive real numbers. The following notation when dealing with sets is often used: x B : x belongs to set B x / B : x not in set B A B : A is a subset of B , i.e. B contains set A A = B : set A equals set B , (if and only if A and B have exactly the same elements : the universe set: contains all elements { } or : the null set: contains no elements Operations on Sets: 4 CHAPTER 2. MATHEMATICAL PRELIMINARIES 5 1. Union: A B { ; A or B } 2. Intersection: A B { ; A and B } 3. Complement: A { ; / A } Definition 2 (disjoint sets) Two sets A and B are disjoint, or mutually exclusive if A B = (i.e. they have no elements in common) De Morgans Laws, given below, are important in dealing with sets (overbar denotes complement): 1. A B = A B 2. A B = A B Definition 3 (algebra of sets) A non-empty collection of subsets = { A 1 ,A 2 , } of a set , ( A i ) is called an algebra of sets if:...
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## This note was uploaded on 08/06/2008 for the course ECE 342 taught by Professor Li during the Fall '05 term at Lehigh University .

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note01_probability - Contents 2 Mathematical Preliminaries...

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