lecture03 - ECE 5510: Random Processes Lecture Notes Fall...

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Unformatted text preview: ECE 5510: Random Processes Lecture Notes Fall 2008 Lecture 3 Today: (1) Independence (2) Conditional Probability Announcements: 1. HW 1 due Thursday at 5pm. It is a short HW, dont get behind. 2. Reading for today: Sections 1.5,1.6. 3. Reading for Thursday: 1.7-1.10, and start into Chapter 2. 4. RSS Data is online. Please start Application Assignment 1. 0.1 Independence Defn: Independence of a Pair of Sets Sets A and B are independent if P [ A B ] = P [ A ] P [ B ]. Defn: Independence of a Sequence of Sets Sets A 1 ,... ,A n are independent if P [ A i A j ] = P [ A i ] P [ A j ] , i negationslash = j, AND P [ A i A j A k ] = P [ A i ] P [ A j ] P [ A k ] , i negationslash = j negationslash = k, AND so on, AND finally P [ A 1 A 2 A n ] = P [ A i ] P [ A j ] P [ A n ] , That is, each pair, triplet, quadruplet, and so on, obeys this product rule. Example: Sets that are pairwise independent but not mu- tually independent Consider S = { 1 , 2 , 3 , 4 } with equal probability, and events A = { 1 , 2 } , B = { 1 , 3 } , C = { 1 , 4 } . 1. Are A and B independent? P [ { 1 } ] = 1 / 4 = (1 / 2)(1 / 2). Yep. 2. Are B and C independent? P [ { 1 } ] = 1 / 4 = (1 / 2)(1 / 2). Yep. ECE 5510 Fall 2008 2 3. Are A , B , and C independent? P [ { 1 } ] = 1 / 4 negationslash = (1 / 2)(1 / 2)(1 / 2)....
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lecture03 - ECE 5510: Random Processes Lecture Notes Fall...

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