This preview shows pages 1–3. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: 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 afternoonWed night, and to a startup in San Diego for research collaboration all Friday. My office hours for next week: Mon 8:009:30 am, Thu 1:003:00 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 didnt previously specify 5pm.) Application Assignment 2 is now posted and will be due Thu Sept 25. Reading for today: 3.13.5, 3.7. Reading for Tuesday: t.b.a. 1 Continuous Random Variables Defn: 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 } ] = > 0. Let N = 1 + 1. (Eg., = 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 = N > 1 . Contradiction! Thus P [ { x } ] = 0 , x S . However, CDFs are still meaningful. 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 ]] = , x < x, 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 ]] = , x < a x a b a , a x < b 1 , x b 1.2 Probability Density Function (pdf) Defn: Probability density function (pdf) The pdf of a continuous r.v.The pdf of a continuous r....
View
Full
Document
 Fall '08
 Chen,R

Click to edit the document details