RP-HW1 - Kyung Hee University Department of Electronics and Radio Engineering C1002900 Random Processing Homework 1 Spring 2010 Professor Hyundong

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1 Kyung Hee University Department of Electronics and Radio Engineering C1002900 Random Processing Homework 1 Spring 2010 Professor Hyundong Shin Issued: March 22, 2010 Due: March 31, 2010 Reading: Course textbook Chapters 1–6 HW 1.1 A random variable X has a cumulative distribution function (CDF)   ,. x X Fx e x  2 10 (a) Calculate the following probabilities:  . PX 1 2 2 (b) Find the probability density function (PDF)   X f x of X . (c) Let Y be a random variable obtained from X as follows : , X Y X 02 12 Find the PDF   Y fy of Y . HW 1.2 Let X and Y be independent identically distributed (i.i.d.) random variables with common density function  , , f   1 0o t h e r w i s e . Let SXY  . (a) Find and sketch   S f s . (b) Find and sketch   XS f xs versus x with s viewed as a known parameter.
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2 (c) The conditional mean of X given Ss is   XS E XS s x f xsdx   . Find . EXS 05 . (d) The conditional mean of X given S is E XS x f xSdx   . Since X S is a function of the random variable S , it is also a random variable. Find the density function of X S . HW 1.3 Wanting to browse the net, Bob uses his high-speed 300 -baud modem to connect through his Internet Service Provider. The modem transmits bits in such a fashion that 1 is sent if a given bit is zero and 1 is sent if a given bit is one. The telephone line has an additive zero-mean Gaussian noise with variance 2 (so, the receiver on the other end gets a signal which is the sum of the transmitted signal and the channel noise). The value of the noise is assumed to be independent of the encoded signal value. , 2 0 We assume that the probability of the modem sending 1 is p and the probability of sending p 1 . (a) Suppose we conclude that an encoded signal 1 was sent when the value received on the other end of the line is less than a (where a   11 ), and conclude 1 was sent when the value is more than a . What is the probability of making an error? (b) Answer part (a) assuming that / p 25 , / a 12 , and /  2 14 . HW 1.4 You take a safari trip to the Porobilati game reserve. A highlight of the game reserve is the Po- seni river where one can watch deer and elephants coming to drink water. Deer come to the river according to a Poisson distribution with arrival rate d 
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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-HW1 - Kyung Hee University Department of Electronics and Radio Engineering C1002900 Random Processing Homework 1 Spring 2010 Professor Hyundong

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