# FL2008ECE757_Additional Problems - 3 Find the values of...

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UNIVERSITY OF NEW HAMPSHIRE DEPARTMENT OF ELECTRICAL AND COMPUTER ENGINEERING ECE757 - Commmunication Systems FALL 2008 Additional Problems: 1. A rectangle has dimensions that are random variables. The base, X, is a random variable uniformly distributed from 0 to 5. The height, Y, is a random variable uniformly distributed from 0 to X. Find the expected value of the area of the rectangle. 2. In a binary communication system (shown below), a 0 or 1 is transmitted. Because of channel noise, a 0 can be received as a 1 and vice versa. Let m 0 and m 1 denote the events of transmitting 0 and 1, respectively. Let r 0 and r 1 denote the events of receiving 0 and 1, respectively. Let P[m 0 ]=0.5, P[r 1 |m 0 ]=p=0.1, and P[r 0 |m 1 ]=q=0.2 m 0 m 1 P[m 1 ] P[m 0 ] P[r 1 |m 1 ] P[r 0 |m 0 ] P[r 1 |m 0 ] P[r 0 |m 1 ] r 0 r 1 a) Find P[r ] and P[r ] 0 1 b) If a 0 was received, what is the probability that a 0 was sent? c) If a 1 was received, what is the probability that a 1 was sent? d) Calculate the Probability of error P e e) Calculate the probability that the transmitted signal is correctly read at the receiver.

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Unformatted text preview: 3. Find the values of constants a and b such that: F X (x) = [1 - aEXP(-x/b)]u(x) Is a valid distribution function {note that u(x) is the unit step function} 4. The pdf of a random variable X is given by: ECE757-Practice Problems ⎩ ⎨ ⎧ ≤ ≤ = otherwise b x a k x f X ) ( Where, k, is a constant. a) Determine the value of k. b) Let a = -1 and b = 2. Calculate P[|X| ≤ c] for c=1/2 5. Let X and Y be defined by: X=cos( Φ ) and Y=sin( Φ ), where Φ is a random variable uniformly distributed over [0, 2 π ]. a) Show that X and Y are uncorrelated b) Show that X and Y are not independent. 6. Consider a random process X(t) given by: X(t)=Acos( ω t+ Φ ), Where, A and ω are constants and Φ is a uniform random variable over [-π , π ]. Show that X(t) is Wide Sense Stationary. ECE757-Practice Problems...
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FL2008ECE757_Additional Problems - 3 Find the values of...

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