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# lecture10 - ECE 5510 Random Processes Lecture Notes Fall...

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ECE 5510: Random Processes Lecture Notes Fall 2008 Lecture 10 Today: (1) Conditional Joint pmfs and pdfs Exam 1 Return: average = 92.7, Exam 1 stdev = 9.9. HW 5 due Tuesday October 10 at 6pm in the HW locker. Reading for today: Y&G 4.8-4.10, 5.4 Reading for Tue, Oct 7: Y&G 4.7, 4.11, 5.6, 5.7 1 Review of Joint Distributions This is Sections 4.1-4.5. For two random variables X 1 and X 2 , Joint CDF: F X 1 ,X 2 ( x 1 ,x 2 ) = P [ { X 1 x 1 } ∩ { X 2 x 2 } ] It is the probability that both events happen simultaneously. Joint pmf: P X 1 ,X 2 ( x 1 ,x 2 ) = P [ { X 1 = x 1 } ∩ { X 2 = x 2 } ] It is the probability that both events happen simultaneously. Joint pdf: f X 1 ,X 2 ( x 1 ,x 2 ) = 2 ∂x 1 ∂x 2 F X 1 ,X 2 ( x 1 ,x 2 ) The pmf and pdf still integrate/sum to one, and are non-negative. Now, to find a probability, you must double sum or double integrate. For example, for event B S , Discrete case: P [ B ] = ∑ ∑ ( X 1 ,X 2 ) B P X 1 ,X 2 ( x 1 ,x 2 ) Continuous Case: P [ B ] = integraltext integraltext ( X 1 ,X 2 ) B f X 1 ,X 2 ( x 1 ,x 2 ) We also talked about marginal distributions: Marginal pmf: P X 2 ( x 2 ) = X 1 S X 1 P X 1 ,X 2 ( x 1 ,x 2 ) Marginal pdf: f X 2 ( x 2 ) = integraltext X 1 S X 1 f X 1 ,X 2 ( x 1 ,x 2 ) Finally we talked about independence of random variables.

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lecture10 - ECE 5510 Random Processes Lecture Notes Fall...

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