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lecture09

# lecture09 - ECE 5510 Random Processes Lecture Notes Fall...

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ECE 5510: Random Processes Lecture Notes Fall 2009 Lecture 9 Today: (1) Joint distributions: Intro Exam is a week from today in class (Sept. 29) Thursday Sept 24 is a review session. Bring questions to go over. HW 4 due Sept 24 at 10:45 am (at start of lecture). No late HW is accepted, as we will hand out solutions in class. OH are today until 1:15. Extra OH tomorrow 9:30-11am. Appl Assignment 2 is due Sept 24 at midnight. Discussion? Option: Record lectures into 5 minute video segments, for watching prior to lecture. Then, the extra time in class would be used to answer questions and do examples. Would you watch? 1 Joint distributions: Intro (Multiple Ran- dom Variables) Often engineering problems can’t be described with just one random variable. And random variables are often related to each other. For example: 1. ICs are made up of resistances, capacitances, inductances, and transistor characteristics, all of which are random, dependent on the outcome of the manufacturing process. A voltage read- ing at some point in the IC may depend on many of these parameters. 2. A chemical reaction may depend on the concentration of mul- tiple reactants, which may change randomly over time. 3. Control systems for vehicles may measure from many different sensors to determine what to do to control the vehicle.

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ECE 5510 Fall 2009 2 (a) X 2 X 1 x 2 x 1 (b) X 2 X 1 c b d a Figure 1: (a) A 2-D joint CDF gives the probability of ( X 1 ,X 2 ) in the area shown. (b) The smaller area shown can also be calculated from the joint CDF. 1.1 Event Space and Multiple Random Variables Def’n: Multiple Random Variables A set of n random variables which result from the same experiment or measurement are a mapping from the sample space S to R n Example: Two dice are rolled. An outcome s S is the result of the experiment of rolling two dice. Let X 1 ( s ) be the number on die 1, and X 2 ( s ) be the number on die 2. The coordinate ( X 1 ,X 2 ) is a function of s and lies in a two-dimensional space R 2 or more specifically, { 1 , 2 , 3 , 4 , 5 , 6 } 2 .
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