Chpt 4 Random Process - Supplementary Material to Chapter 4...

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1 Supplementary Material to Chapter 4 Random Process
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2 Outline Due to the presence of random noise during transmission, any digital communication system is subject to errors. Such errors are random in nature. Therefore the theory of probability and random process is the mathematical tool to evaluate system performance. The following is to review some basic concepts. Knowledge of probability is assumed from your mathematical courses. We will discuss both discrete and continuous definitions of probabilities. More examples will be given in the tutorials. You are highly recommended to read the textbook for an in-depth discussion.
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3 A Brief Review of Probability Imaging that we pick up a colored ball from a black box. We try it for N times. Denote by n the times that we get a red ball. Then the probability of getting a red ball is loosely defined as Observations: (1) Probability is never negative. (2) Probability is always no larger than 1. (3) The chance to get a red ball in the next try is roughly n / N . However, this does not guarantee you “get” or “not get” a red ball in the next try. Pr{getting a red ball in each try} lim{ } N n N →∞ =
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4 Continuous Probability The above definition is not convenient for a continuous random variable. In this case, the following expression is more meaningful Example: Suppose you are throwing a ball. The probability that the ball drops at a distance of exactly 10m is zero. It is more meaningful to say the probability of dropping point between 9.5m and 10m. Cumulative distribution function A new definition is now introduced to overcome the near zero probability difficulty. Let X be a random variable, its cumulative distribution function (CDF) is P ( x ) = Pr{ X x } Clearly Pr{ a < X b } = P ( b )- P ( a ) Pr{ } aXb < 01 0 m
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5 Probability Density Function (PDF) The PDF of X , denoted by p ( x ), is defined by, The physical meaning of p ( x ) is that p ( x ) Δ x is the probability of X falling in ( x , Δ x ]. Ques.: (a) What is the expression for Pr{ X x }? (b) What is the expression for Pr{ X > x }? Pr{ } ( ) b a aXb p v d v <≤= xx + x p ( x )
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6 The Relationship between PDF and CDF PDF and CDF form a differentiation/integration pair. We can modify the expression for mean value, mean power and variance as Ques: (1) Can p ( x )>1?
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Chpt 4 Random Process - Supplementary Material to Chapter 4...

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