chapter_4_probability_review - Chapter 4 Random Processes...

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Chapter 4: Random Processes Dr. Barış Atakan 1
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Random Variables Random Variables: A random variable is a mapping from the sample space Ω to the set of real numbers. Random variables are denoted by capital letters X, Y, etc .; individual values of the random variable X are X(ω). A random variable is discrete if the range of its values is finite. This range is usually denoted by {x i }. 2
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Random Variables The cumulative distribution function (CDF) of a random variable X is defined as 3 or ) ( ) ( ) ( . 8 lim where ) ( ) ( . 7 ) ( 1 ) ( . 6 ) ( ) ( ) ( . 5 lim , ) ( ) ( i.e., right, the from continuous is ) ( . 4 1 ) ( lim and 0 ) ( lim 3. ing nondecreas is ) ( . 2 1 ) ( 0 . 1 0 0 0 0 x x   a F a F a X P b b b F b X P a F a X P a F b F b X a P a a a F a F x F x F x F x F x F X X X X X X X X X X X X X
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Random Variables For discrete random variables F X (x) is a stair- case function A random variable is called continuous if F X (x) is a continuous function 4
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Random Variables The probability density function (PDF) of a random variable X is defined as the derivative of F X (x); i.e., In case of discrete random variables , the PDF involves impulses 5 For discrete random variables , it is more common to define the probability mass function (PMF), which is defined as {p i } where p i =P(X = x i ). Obviously for all i we have p i ≥ 0 and ∑ i p i = 1 x X X b a X X X du u f x F b X a P dx x f dx x f f ) ( ) ( . 4 ) ( ) ( . 3 1 ) ( . 2 0 . 1 -
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Random Variables: Important Random Variables Bernoulli Random Variable: A discrete random variable taking two values one and zero with probabilities p and 1 − p
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