EE558 - Digital Communications Lecture 3: Review of Probability and Random Processes Dr. Duy Nguyen
Outline Introduction 1 Probability and Random Variables 2 Random Processes 3 Introduction 2
Introduction The main objective of a communication system is the transfer of information over a channel. Message signal is best modeled by a random signal Two types of imperfections in a communication channel: ◮ Deterministic imperfection, such as linear and nonlinear distortions, inter-symbol interference, etc. ◮ Nondeterministic imperfection, such as addition of noise, interference, multipath fading, etc. We are concerned with the methods used to describe and characterize a random signal, generally referred to as a random process (also commonly called stochastic process). In essence, a random process is a random variable evolving in time. Introduction 3
Outline Introduction 1 Probability and Random Variables 2 Random Processes 3 Probability and Random Variables 4
Sample Space and Probability Random experiment : its outcome, for some reason, cannot be predicted with certainty. Examples: throwing a die, flipping a coin and drawing a card from a deck. Sample space : the set of all possible outcomes, denoted by Ω . Outcomes are denoted by ω ’s and each ω lies in Ω , i.e., ω ∈ Ω . A sample space can be discrete or continuous . Events are subsets of the sample space for which measures of their occurrences, called probabilities, can be defined or determined. Probability and Random Variables 5
Example of Throwing a Fair Die Ω Various events can be defined: “the outcome is even number of dots”, “the outcome is smaller than 4 dots”, “the outcome is more than 3 dots”, etc. Probability and Random Variables 6
Three Axioms of Probability For a discrete sample space Ω , define a probability measure P on Ω as a set function that assigns nonnegative values to all events, denoted by E , in Ω such that the following conditions are satisfied Axiom 1: 0 ≤ P ( E ) ≤ 1 for all E ∈ Ω (on a % scale probability ranges from 0 to 100%. Despite popular sports lore, it is impossible to give more than 100%). Axiom 2: P (Ω) = 1 (when an experiment is conducted there has to be an outcome). Axiom 3: For mutually exclusive events 1 E 1 , E 2 , E 3 ,. . . we have P ( � ∞ i =1 E i ) = � ∞ i =1 P ( E i ) . 1 The events E 1 , E 2 , E 3 ,. . . are mutually exclusive if E i ∩ E j = ⊘ for all i � = j , where ⊘ is the null set. Probability and Random Variables 7
Important Properties of the Probability Measure 1. P ( E c ) = 1 − P ( E ) , where E c denotes the complement of E . This property implies that P ( E c ) + P ( E ) = 1 , i.e., something has to happen. 2. P ( ⊘ ) = 0 (again, something has to happen). 3. P ( E 1 ∪ E 2 ) = P ( E 1 ) + P ( E 2 ) − P ( E 1 ∩ E 2 ) . Note that if two events E 1 and E 2 are mutually exclusive then P ( E 1 ∪ E 2 ) = P ( E 1 ) + P ( E 2 ) , otherwise the nonzero common probability P ( E 1 ∩ E 2 ) needs to be subtracted off. 4. If E 1 ⊆ E 2 then P ( E 1 ) ≤ P ( E 2 ) . This says that if event E 1 is contained in E 2 then occurrence of E 1 means E 2 has occurred but the converse is not true. Probability and Random Variables 8
Conditional Probability We observe or are told that event E 1 has occurred but are actually interested in event E 2 : Knowledge that of E 1 has occurred changes the probability of E 2 occurring. If it was P ( E 2 ) before, it now becomes P ( E 2 | E 1 ) , the probability of E 2 occurring given that event E 1 has occurred. This conditional probability is given by � P ( E 2 ∩ E 1 ) , if P ( E 1 ) � = 0 P ( E 1 ) P ( E 2 | E 1 ) = . 0 , otherwise If P ( E 2 | E 1 ) = P ( E 2 ) , or P ( E 2 ∩ E 1 ) = P ( E 1 ) P ( E 2 ) , then E 1 and E 2 are said to be statistically independent . Bayes ’ rule P ( E 2 | E 1 ) = P ( E 1 | E 2 ) P ( E 2 ) , P ( E 1 ) Probability and Random Variables 9
Total Probability Theorem The events { E i } n i =1 partition the sample space Ω if: n � (i) E i = Ω (1) i =1 (ii) E i ∩ E j = ⊘ for all 1 ≤ i, j ≤ n and i � = j (2) If for an event A we have the conditional probabilities { P ( A | E i ) } n i =1 , P ( A ) can be obtained as n � P ( A ) = P ( E i ) P ( A | E i ) . i =1 Bayes’ rule : P ( E i | A ) = P ( A | E i ) P ( E i ) P ( A | E i ) P ( E i ) = j =1 P ( A | E j ) P ( E j ) . � n P ( A ) Probability and Random Variables 10
Random Variables Ω ω ω 1 2 ω 4 ω 3 R ( ω ) ( ω ) ( ω ) ( ω ) x x x x 1 3 2 4 A random variable is a mapping from the sample space Ω to the set of real numbers. We shall denote random variables by boldface, i.e., x , y , etc., while individual or specific values of the mapping x are denoted by x ( ω ) . Probability and Random Variables 11
Random Variable in the Example of Throwing a Fair Die Ω R 5 1 2 3 4 6 There could be many other random variables defined to describe the outcome of this random experiment! Probability and Random Variables 12
Cumulative Distribution Function (cdf) cdf gives a complete description of the random variable. It is defined as: F x ( x ) = P ( ω ∈ Ω : x ( ω ) ≤ x ) = P ( x ≤ x ) . The cdf has the following properties: 1. 0 ≤ F x ( x ) ≤ 1 2. F x ( x ) is nondecreasing: F x ( x 1 ) ≤ F x ( x 2 ) if x 1 ≤ x 2 3. F x ( −∞ ) = 0 and F x (+ ∞ ) = 1 4. P ( a < x ≤ b ) = F x ( b ) − F x ( a ) . Probability and Random Variables 13
Typical Plots of cdf I A random variable can be discrete , continuous or mixed . F x ( ) x 1 . 0 (a) x 0 − ∞ ∞ Probability and Random Variables 14
Typical Plots of cdf II ( ) F x x 1 . 0 (b) x 0 − ∞ ∞ F x ( ) x 1 . 0 (c) x 0 − ∞ ∞ Probability and Random Variables 15
Probability Density Function (pdf) The pdf is defined as the derivative of the cdf: f x ( x ) = d F x ( x ) . d x It follows that: P ( x 1 ≤ x ≤ x 2 ) = P ( x ≤ x 2 ) − P ( x ≤ x 1 ) � x 2 = F x ( x 2 ) − F x ( x 1 ) = f x ( x )d x. x 1 Basic properties of pdf: 1. f x ( x ) ≥ 0 . � ∞ 2. −∞ f x ( x )d x = 1 . � 3. In general, P ( x ∈ A ) = A f x ( x )d x . For discrete random variables, it is more common to define the probability mass function (pmf): p i = P ( x = x i ) . Note that, for all i , one has p i ≥ 0 and � i p i = 1 . Probability and Random Variables 16
Bernoulli Random Variable f ( ) x F x ( ) x x 1 1 p − ( ) p (1 p ) − x x 0 1 0 1 A discrete random variable that takes two values 1 and 0 with probabilities p and 1 − p . Good model for a binary data source whose output is 1 or 0. Can also be used to model the channel errors. Probability and Random Variables 17
Binomial Random Variable f ( ) x x 0 . 30 0 . 25 0 . 20 0 . 15 0 . 10 0 . 05 x 0 2 4 6 A discrete random variable that gives the number of 1’s in a sequence of n independent Bernoulli trials. n � n � � n � n ! � p k (1 − p ) n − k δ ( x − k ) , where f x ( x ) = = k !( n − k )! . k k k =0 Probability and Random Variables 18
Uniform Random Variable f ( ) x F x ( ) x x 1 1 b − a x x a 0 b a 0 b A continuous random variable that takes values between a and b with equal probabilities over intervals of equal length. The phase of a received sinusoidal carrier is usually modeled as a uniform random variable between 0 and 2 π . Quantization error is also typically modeled as uniform. Probability and Random Variables 19
Gaussian (or Normal) Random Variable f ( ) x F x ( ) x x 1 2 2 πσ 1 1 2 x x 0 µ 0 µ A continuous random variable whose pdf is: − ( x − µ ) 2 � � 1 √ f x ( x ) = 2 πσ 2 exp , 2 σ 2 µ and σ 2 are parameters. Usually denoted as N ( µ, σ 2 ) . Most important and frequently encountered random variable in communications. Probability and Random Variables 20
Functions of A Random Variable The function y = g ( x ) is itself a random variable. From the definition, the cdf of y can be written as F y ( y ) = P ( ω ∈ Ω : g ( x ( ω )) ≤ y ) . Assume that for all y , the equation g ( x ) = y has a countable number of solutions and at each solution point, d g ( x ) / d x exists and is nonzero. Then the pdf of y = g ( x ) is: f x ( x i ) � f y ( y ) = , � � � d g ( x ) � � i � � d x � � x = x i � � where { x i } are the solutions of g ( x ) = y . A linear function of a Gaussian random variable is itself a Gaussian random variable . Probability and Random Variables 21
Expectation of Random Variables I Statistical averages , or moments , play an important role in the characterization of the random variable. The expected value (also called the mean value, first moment) of the random variable x is defined as � ∞ m x = E { x } ≡ xf x ( x )d x, −∞ where E denotes the statistical expectation operator . In general, the n th moment of x is defined as � ∞ E { x n } ≡ x n f x ( x )d x. −∞ Probability and Random Variables 22
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