RP-Midterm Exam-sol

# RP-Midterm Exam-sol - Kyung Hee University Department of...

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1 Kyung Hee University Department of Electronics and Radio Engineering C1002900 Random Processing Midterm Exam Solutions Spring 2010 Professor Hyundong Shin April 26, 2010 19:00 – 21:30 (150 minutes) This is an in-class closed book exam, but one A4-size sheet of notes (single side) is allowed. A correct answer does not guarantee full credit, and a wrong answer does not guarantee loss of credit. You should clearly but concisely expose your reasoning and show all rele- vant work . Your grade on each problem will be based on our best assessment of your level of understanding as reflected by what you have written in the answer sheets. Please be neat we cannot grade what we cannot decipher! Problem 1 (10 points) Arrivals of certain events at points in time are known to constitute a Poisson random variable, but it is unknown which of two possible values of the average arrival rate describes the ran- dom variable. Our a priori estimate is that  2 or 4 with equal probability. We observe the random variable for t units of time and observe exactly k arrivals. Given this information, de- termine the conditional probability that 2 . Check to see whether or not your answer is rea- sonable for some simple limiting values for k and t . Solution: We have a Poisson random variable with an average arrival rate , which is equally likely to be either 2 or 4 . Thus,  . 1 24 2  We observe the variable for t time units and observe k arrivals. The conditional probability that 2 is, by definition . kt 2 and arrivals in time 2 arrivals in time arrivals in time Now, we know that  . ! k t te k  2 2 and arrivals in time arrivals in time 2 2 2 1 2

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2 Similarly,    . ! k t te kt k  4 4 1 4 and arrivals in time 2 Hence,      ! !! . k t kk tt k t k e  2 24 2 2 2 2 2 arrivals in time 22 2 1 12 To check whether this answer is reasonable, suppose t is large and 2 (observed arrival rate equals 2 ). Then, 2 arrivals in time approaches 1 as t goes to . Similarly, if t is large and 4 (observed arrival rate equals 4 ), then 2 arrivals in time approach- es 0 as t goes to . Problem 2 (15 points) (a) (5 points) Let   , X 01 and   , Y be statistical independent. Consider a complex random variable Z Xj Y  . Find the probability that the instantaneous power Z 2 of Z is less than or equal to its average power. (b) (5 points) Let   , X and   , Y be statistical independent. 1 Consider random variables Z XY and WXY  . Are Z and W independent? Uncorrelated? (c) (5 points) Let U and V have the joint PDF , ,, , ,o t h e r w i s e . UV uv u v fu v  0 Let XU 2 and   YU V 1 . Find the joint PDF   , , XY f x y of X and Y .
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## This note was uploaded on 06/10/2010 for the course ELECTRONIC C1002900 taught by Professor Hyungdongshin during the Spring '10 term at Kyung Hee.

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RP-Midterm Exam-sol - Kyung Hee University Department of...

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