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

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ECE 5510: Random Processes Lecture Notes Fall 2008 Lecture 6 Today: (1) Cts r.v.s, (2) Expectation of Cts r.v.s, (3) Method of Moments My travel schedule next week: I will be going to a conference in San Francisco Mon afternoon-Wed night, and to a startup in San Diego for research collaboration all Friday. My office hours for next week: Mon 8:00-9:30 am, Thu 1:00-3:00 pm. Use email to reach me otherwise. Lecture 7 on Tuesday, Sept. 16 will be taught by Dr. Chen. I will give the Thursday, Sept. 18th lecture. HW 2 due today at 5pm. HW 3 assigned today. Application Assignment 1 due this coming Tuesday (at mid- night, because I didn’t previously specify 5pm.) Application Assignment 2 is now posted and will be due Thu Sept 25. Reading for today: 3.1-3.5, 3.7. Reading for Tuesday: t.b.a. 1 Continuous Random Variables Def’n: Continuous r.v. A r.v. is continuous if its range S X is uncountably infinite ( i.e. not countable). Eg, the ‘wheel of fortune’, for which X [0 , 1). Probability mass functions 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 = 1 ǫ + 1. (Eg., ǫ = 0 . 001 N = 1001). Then P bracketleftBigg N - 1 uniondisplay n =0 { n N } bracketrightBigg = N - 1 summationdisplay n =0 P bracketleftBig { n N } bracketrightBig = N - 1 summationdisplay n =0 ǫ = Nǫ > 1 . Contradiction! Thus P [ { x } ] = 0 , x S . However, CDFs are still meaningful.
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ECE 5510 Fall 2008 2 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) 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 ]] = 0 , x < 0 x, 0 x < 1 1 , x 1 In general, for a uniform random variable X that has S X = [ a, b ), F X ( x ) = P [[ a, x ]] = 0 , 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 ( a ) is the density . Not a probability! .
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