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Chapt-02_Slides

# Chapt-02_Slides - Chapter 2 Probability Statistics and...

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1 Chapter 2 Probability, Statistics, and Traffic Theories Copyright © 2011, Dr. Dharma P. Agrawal and Dr. Qing-An Zeng. All rights reserved. 1 Outline Introduction Probability Theory and Statistics Theory Random variable Random variables Probability mass function (pmf) Probability density function (pdf) Cumulative distribution function (CDF) Expected value, n th moment, n th central moment, and variance Some important distributions Traffic Theory Poisson arrival model etc Copyright © 2011, Dr. Dharma P. Agrawal and Dr. Qing-An Zeng. All rights reserved. 2 Poisson arrival model, etc. Basic Queuing Systems Little’s law Basic queuing models

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2 Introduction Several factors influence the performance of wireless systems: Density of mobile users Cell size Moving direction and speed of users (Mobility models) Call rate, call duration Interference, etc. Probabilit statistics theor and traffic patterns Copyright © 2011, Dr. Dharma P. Agrawal and Dr. Qing-An Zeng. All rights reserved. 3 Probability, statistics theory and traffic patterns, help make these factors tractable Probability Theory and Statistics Theory Random Variables (RVs) Let S be sample associated with experiment E X is a function that associates a real number to each s S RVs can be of two types: Discrete or Continuous Discrete random variable => probability mass function (pmf) Continuous random variable => probability density function (pdf) R Copyright © 2011, Dr. Dharma P. Agrawal and Dr. Qing-An Zeng. All rights reserved. 4 s X(s) X S R S
3 Discrete Random Variables In this case, X(s) contains a finite or infinite number of values The possible values of X can be enumerated E.g., throw a 6 sided dice and calculate the probability of a particular number appearing. 0.3 02 Probability Copyright © 2011, Dr. Dharma P. Agrawal and Dr. Qing-An Zeng. All rights reserved. 5 1 2 34 6 0.1 0.1 0.1 0.2 0.2 5 Number Discrete Random Variables The probability mass function (pmf) p(k) of X is defined as defined as: p(k) = p(X = k), for k = 0, 1, 2, . .. where 1 . Probability of each state occurring 0 p(k) 1, for every k; Copyright © 2011, Dr. Dharma P. Agrawal and Dr. Qing-An Zeng. All rights reserved. 6 2 . Sum of all states p(k) = 1, for all k

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4 Continuous Random Variables In this case, X contains an infinite number of values Mathematically, X is a continuous random variable if there is a function f , called probability density function (pdf) of X that satisfies the following criteria: 1 f(x 0 for all x Copyright © 2011, Dr. Dharma P. Agrawal and Dr. Qing-An Zeng. All rights reserved. 7 1. f(x) 0, for all x; 2. f(x)dx = 1 Cumulative Distribution Function Applies to all random variables A cumulative distribution function cd ) is defined A cumulative distribution function (cdf) is defined as: For discrete random variables: P(k) = P(X k) = P(X = k) all k Copyright © 2011, Dr. Dharma P. Agrawal and Dr. Qing-An Zeng. All rights reserved. 8 For continuous random variables: F(x) = P(X x) = f(x)dx - x
5

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Chapt-02_Slides - Chapter 2 Probability Statistics and...

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