[RP]Lecture Note II

# [RP]Lecture Note II - Kyung Hee University Department of Electronics and Radio Engineering Random Processing Prof Hyundong Shin C1002900 RP Lecture

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Kyung Hee University Random Processing Department of Electronics and Radio Engineering Prof. Hyundong Shin Communications and Coding Theory Laboratory (CCTLAB) II-1 C1002900 RP Lecture Handout II: Random Variables and Their Distributions Reading: Chapter 4 2.1 Random Variable Formally, a random variable X is a real valued function on the sample space . Definition 2.1 (Random Variable): Let a probability space  ,, P be given. A random variable is a function X from to the real line , which is -measurable, meaning that for any number x ,   : Xx  . If is finite or countable infinite, then can be the set of all subsets of , in which case real valued function on is a random variable. If P is given as in the uni- form phase example with equal to the Borel subsets of   , 0 2 , then the random va- riables on P are called the Borel measurable function on . 2.2. Distribution and Density Function Definition 2.2 (CDF): The cumulative distribution function (CDF) of a random varia- ble X , denoted by X F , is defined by      : (for shorthand notation). X Fx P X x PX x  This distribution completely characterizes the random variable.

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Kyung Hee University Random Processing Department of Electronics and Radio Engineering Prof. Hyundong Shin Communications and Coding Theory Laboratory (CCTLAB) II-2 If X denotes the outcome of rolling a fair die and if Y is uniformly distributed on the interval   , 01, then X F and Y F are shown in Figure 2.1. X F 1 1 0 2 3 4 5 6 1 1 Y F 0 Figure 2.1: Examples of CDFs. Example 2.1: -1 0 1   X F x x 1) What is the random variable X ? 2)   ? PX  0 Proposition 2.1: A function F is the CDF of some random variable if and only if it has the following three properties: F.1 F is nondecreasing. F.2  lim x Fx  1 and lim x  0. F.3 F is right continuous, that is   XX F xF x .
Kyung Hee University Random Processing Department of Electronics and Radio Engineering Prof. Hyundong Shin Communications and Coding Theory Laboratory (CCTLAB) II-3          : definedtobethesizeofthejumpof at . XX X X PX x PX x PX x Fx Fx F xF x   The CDF of a random variable X determines   P Xx for any real number x . How- ever, what about   P and   P ? Let ,, xx 12 be a monotone nondecreas- ing sequence that converges to x from the left. This means ij xxx  for and lim jj  . Then, the events   j are nested:     i j  for , and the union of all such events is the event   . Therefore,     lim lim . i i Xi i X P PXx Fx  Hence,      X P Xx Fx Fx where X F x is defined to be the size of the jump of X F at x . For example, if X has the CDF shown in Example 2.1, then   / PX 0 1 2 . The requirement that X F is right

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## This note was uploaded on 06/10/2010 for the course ELECTRONIC C1002900 taught by Professor Hyungdongshin during the Spring '10 term at Kyung Hee.

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[RP]Lecture Note II - Kyung Hee University Department of Electronics and Radio Engineering Random Processing Prof Hyundong Shin C1002900 RP Lecture

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