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| + | [[ECE_PhD_Qualifying_Exams|ECE Ph.D. Qualifying Exam]] | ||
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| + | Communication, Networking, Signal and Image Processing (CS) | ||
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| + | Question 1: Probability and Random Processes | ||
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| + | August 2001 | ||
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Revision as of 01:31, 9 March 2015
Communication, Networking, Signal and Image Processing (CS)
Question 1: Probability and Random Processes
August 2001
2. (25 Points)
Let $ \mathbf{X} $ and $ \mathbf{Y} $ be independent Poisson random variables with mean $ \lambda $ and $ \mu $ , respectively. Let $ \mathbf{Z} $ be a new random variable defined as $ \mathbf{Z}=\mathbf{X}+\mathbf{Y}. $
(a) Find the probability mass function (pmf) of $ \mathbf{Z} $ .
(b)Find the conditional probability mass function (pmf) of $ \mathbf{X} $ conditional on the event $ \left\{ \mathbf{Z}=n\right\} $ . Identify the type of pmf that this is, and fully specify its parameters.
Note
This problem is identical to the example: Addition of two independent Poisson random variables.
