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lecture06 - ECE 5510: Random Processes Lecture Notes Fall...

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Unformatted text preview: ECE 5510: Random Processes Lecture Notes Fall 2009 Lecture 6 Today: (1) Cts r.v.s, (2) Expectation of Cts r.v.s, (3) Method of Moments • HW 2 due today at 5pm. HW 3 assigned today, due Thu, Sept. 17. Application Assignment 1 due Tuesday (at mid- night). • Reading for today: 3.1-3.5, 2.6, 3.7. Reading for Tuesday: S. Kay, “Intuitive Probability and Random Processes using Matlab”, Section 10.7, 8 pages. On WebCT. 1 Continuous Random Variables Def’n: Continuous r.v. A r.v. is continuous-valued if its range S X is uncountably infinite ( i.e. , not countable). E.g. , the ‘Wheel of Fortune’, for which X ∈ [0 , 1). pmfs are meaningless . Why? Because P [ X = x ] = 0. Why is that? Lemma: Let x ∈ [0 , 1). (Eg., x = 0 . 5). Then P [ { x } ] = 0. Proof: Proof by contradiction. Suppose P [ { x } ] = ǫ > 0. Let N = ceilingleftbig 1 ǫ ceilingrightbig + 1. (Eg., ǫ = 0 . 001 → N = 1001). Then P bracketleftBigg N- 1 uniondisplay n =0 braceleftBig n N bracerightBig bracketrightBigg = N- 1 summationdisplay n =0 P bracketleftBigbraceleftBig n N bracerightBigbracketrightBig = N- 1 summationdisplay n =0 ǫ = Nǫ > 1 . Contradiction! Thus P [ { x } ] = 0 , ∀ x ∈ S . However, CDFs are still meaningful. 1.1 Example CDFs for Continuous r.v.s Example: CDF for the wheel of fortune What is the CDF F X ( x ) = P [[0 ,x ]]? By ‘uniform’ we mean that the probability is proportional to the size of the interval. F X ( x ) = P [[0 ,x ]] = a ( x − 0) ECE 5510 Fall 2009 2 for some constant a . Since we know that lim x → + ∞ F X ( x ) = 1, we know that for x = 1, F X ( x ) = a (1 − 0) = 1. Thus a = 1 and F X ( x ) = P [[0 ,x ]] = , x < x, ≤ x < 1 1 , x ≥ 1 In general a uniform random variable X with S X = [ a,b ) has F X ( x ) = P [[ a,x ]] = , x < a x- a b- a , a ≤ x < b 1 , x ≥ b 1.2 Probability Density Function (pdf) Def’n: Probability density function (pdf) The pdf of a continuous r.v. X , f X ( x ), can be written as the deriva- tive of its CDF: f X ( x ) = ∂F X ( x ) ∂x Properties: 1. f X (...
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This note was uploaded on 09/15/2011 for the course ECE 5510 taught by Professor Chen,r during the Fall '08 term at Utah.

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lecture06 - ECE 5510: Random Processes Lecture Notes Fall...

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