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[RP]Lecture Note II

[RP]Lecture Note II - Kyung Hee University Department of...

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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 , : X x   . 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 F x P X x P X 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 , 0 1 , 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) ? P X 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 F x  1 and   lim x F x  0 . F.3 F is right continuous, that is   X X F x F x .
Kyung Hee University Random Processing Department of Electronics and Radio Engineering Prof. Hyundong Shin Communications and Coding Theory Laboratory (CCTLAB) II-3     : defined to be the size of the jump of at . X X X X P X x P X x P X x F x F x F x F x The CDF of a random variable X determines P X x for any real number x . How- ever, what about P X x and P X x ? Let , , x x 1 2 be a monotone nondecreas- ing sequence that converges to x from the left. This means i j x x x for i j and lim j j x x  . Then, the events j X x are nested: i j X x X x for i j , and the union of all such events is the event X x . Therefore, lim lim .

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[RP]Lecture Note II - Kyung Hee University Department of...

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