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

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ECE 5510: Random Processes Lecture Notes Fall 2008 Lecture 9 Today: (1) Conditional Distribution Review (2) Joint distribu- tions: Intro Exam is a week from today in class (Sept. 30) Thursday Sept 25 is a review session. Bring questions to go over. HW 4 due Sept 25 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 3-4pm. Appl Assignment 2 is due Sept 25 at 5pm. 1 Conditional Distribution Review W is Gaussian with mean 0 and variance σ 2 = 16. Given the event C = { W > 0 } , 1. What is f W | C ( w )? First, since W is zero-mean and f W is symmetric, P [ C ] = 1 / 2. Then, f W | C ( w ) = b 2 1 32 π e - w 2 / (32) , w > 0 0 , o.w. 2. What is E W | C [ W | C ]? E W | C [ W | C ] = i 0 2 w 1 32 π e - w 2 / 32 Making the substitution v = w 2 / 32, dv = 2 w/ 32, E W | C [ W | C ] = 32 32 π i 0 e - v = r 32 π 3. What is Var W | C [ W | C ]? E W | C B W 2 | C ± = i 0 2 w 2 1 32 π e - w 2 / 32
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ECE 5510 Fall 2008 2 Knowing that 2 w 2 32 π e - w 2 / 32 is an even expression, the integral for w > 0 is the same as the integral for w < 0 – half the total. E W | C b W 2 | C B = i -∞ w 2 1 32 π e - w 2 / 32 = Var W [ W ] = 16 . The variance is then Var W | C [ W | C ] = E W | C b W 2 | C B ( E W | C [ W | C ]) 2 = 16 32 π 2 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 di±erent sensors to determine what to do to keep the drive safe, stable, and comfortable. 4. etc.
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lecture09 - ECE 5510: Random Processes Lecture Notes Fall...

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