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| + | [[Category:random variables]] | ||
| + | [[Category:probability]] | ||
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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 | ||
| + | </font size> | ||
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| + | August 2001 | ||
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'''2. (25 Points)''' | '''2. (25 Points)''' | ||
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'''(b)'''Find the conditional probability mass function (pmf) of <math class="inline">\mathbf{X}</math> conditional on the event <math class="inline">\left\{ \mathbf{Z}=n\right\}</math> . Identify the type of pmf that this is, and fully specify its parameters. | '''(b)'''Find the conditional probability mass function (pmf) of <math class="inline">\mathbf{X}</math> conditional on the event <math class="inline">\left\{ \mathbf{Z}=n\right\}</math> . Identify the type of pmf that this is, and fully specify its parameters. | ||
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| + | ==Share and discuss your solutions below.== | ||
| + | ---- | ||
| + | ==Solution 1== | ||
'''Note''' | '''Note''' | ||
This problem is identical to the example: [[ECE 600 Exams Addition of two independent Poisson random variables|Addition of two independent Poisson random variables]]. | This problem is identical to the example: [[ECE 600 Exams Addition of two independent Poisson random variables|Addition of two independent Poisson random variables]]. | ||
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Latest revision as of 16:42, 13 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.
Solution 1
Note
This problem is identical to the example: Addition of two independent Poisson random variables.
