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Unformatted text preview: Lecture Notes 7 Random Processes • Definition • IID Processes • Bernoulli Process ◦ Binomial Counting Process ◦ Interarrival Time Process • Markov Processes • Markov Chains ◦ Classification of States ◦ Steady State Probabilities Corresponding pages from B&T: 271–281, 313–340. EE 178: Random Processes Page 7 – 1 Random Processes • A random process (also called stochastic process ) { X ( t ) : t ∈ T } is an infinite collection of random variables, one for each value of time t ∈ T (or, in some cases distance) • Random processes are used to model random experiments that evolve in time: ◦ Received sequence/waveform at the output of a communication channel ◦ Packet arrival times at a node in a communication network ◦ Thermal noise in a resistor ◦ Scores of an NBA team in consecutive games ◦ Daily price of a stock ◦ Winnings or losses of a gambler ◦ Earth movement around a fault line EE 178: Random Processes Page 7 – 2 Questions Involving Random Processes • Dependencies of the random variables of the process: ◦ How do future received values depend on past received values? ◦ How do future prices of a stock depend on its past values? ◦ How well do past earth movements predict an earthquake? • Long term averages: ◦ What is the proportion of time a queue is empty? ◦ What is the average noise power generated by a resistor? • Extreme or boundary events: ◦ What is the probability that a link in a communication network is congested? ◦ What is the probability that the maximum power in a power distribution line is exceeded? ◦ What is the probability that a gambler will lose all his capital? EE 178: Random Processes Page 7 – 3 Discrete vs. ContinuousTime Processes • The random process { X ( t ) : t ∈ T } is said to be discretetime if the index set T is countably infinite, e.g., { 1 , 2 , . . . } or { . . . , 2 , 1 , , +1 , +2 , . . . } : ◦ The process is simply an infinite sequence of r.v.s X 1 , X 2 , . . . ◦ An outcome of the process is simply a sequence of numbers • The random process { X ( t ) : t ∈ T } is said to be continuoustime if the index set T is a continuous set, e.g., (0 , ∞ ) or (∞ , ∞ ) ◦ The outcomes are random waveforms or random occurances in continuous time • We only discuss discretetime random processes: ◦ IID processes ◦ Bernoulli process and associated processes ◦ Markov processes ◦ Markov chains EE 178: Random Processes Page 7 – 4 IID Processes • A process X 1 , X 2 , . . . is said to be independent and identically distributed (IID, or i.i.d.) if it consists of an infinite sequence of independent and identically distributed random variables • Two important examples: ◦ Bernoulli process: X 1 , X 2 , . . . are i.i.d. Bern( p ) , < p < 1 , r.v.s. Model for random phenomena with binary outcomes, such as: * Sequence of coin flips * Noise sequence in a binary symmetric channel * The occurrence of random events such as packets (1 corresponding to an...
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This note was uploaded on 09/29/2008 for the course EE 178 taught by Professor Hewlett during the Spring '08 term at Stanford.
 Spring '08
 Hewlett

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