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RP-HW1 - Kyung Hee University Department of Electronics and...

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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 F x e x 2 1 0 (a) Calculate the following probabilities: . P X P X P X 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 0 2 1 2 Find the PDF   Y f y of Y . HW 1.2 Let X and Y be independent identically distributed (i.i.d.) random variables with common density function , , f      1 0 1 0 otherwise. Let S X Y . (a) Find and sketch   S f s . (b) Find and sketch X S f x s versus x with s viewed as a known parameter.
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2 (c) The conditional mean of X given S s is X S E X S s xf x s dx  . Find . E X S 0 5 . (d) The conditional mean of X given S is X S X S E X S xf x S dx  . 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 1 is 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 1 1 ), 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 2 5 , / a 1 2 , and / 2 1 4 .
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