HW5_Spring11_sol

# HW5_Spring11_sol - ELEC210 Spring 2011 Homework 5 Solution...

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ELEC210 Spring 2011 Homework 5 Solution 1. Let n X U = where n is a positive integer and U is a uniform random variable in the interval [-1,1]. Find the cdf and pdf of X . Solution: () 1/2, 1 1 0, otherwise u pU u −≤ ≤ == i) When n is odd, 1/ nn X Fx PU x PUx ⎤⎡ =≤ = ⎦⎣ ( ) 1, 0 X xF x <− = 1/ 1/ 11 , 1 1 22 X x x x ⎡⎤ = −− = + ⎣⎦ ( ) 1 X x >= 1 1 1 , 1 1 2 otherwise n X xx fx n = ii) When n is even, -1 -0.5 0 0.5 1 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

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() 1/ 1/ nn n X Fx PU x Px Ux ⎤⎡ =≤ = ⎦⎣ 0, 0 X xF x <= 1/ 1/ 1 01 , 2 2 X x Fx x x ≤≤ = = ( ) 1, 1 X x >= 1 1 1 , 0 1 elsewhere n X xx fx n = 2. The amplitude of a radio signal X is a Rayleigh random variable with pdf: 22 /2 2 0, 0 x X x e x α => > Find the pdf of 3 ZX + =− ( ( ) () m a x0 , XX + = ). Solution: 3 3 3, 3 X + =−= −> [][] [] 2 2 2 3 9/2 2 0 3/ 2 0 3 1 0 33 1 , 0 x z z x PZ z PX e d x e z z PX z e z αα −− −+ < ≤= = = −≤ = ≤+ =− > Therefore, -1 -0.5 0 0.5 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
() 2 9/2 01 Z PZ e α == ( ) 22 3/ 2 2 2 3 1, 0 zz ZZ dd z fz Fz e e z dz dz αα −+ + = > 3. The input X to a communication channel is “-1” or “1”, with respective probabilities 1/3 and 2/3. The output of the channel Y is equal to: the corresponding input X with probability 1 e pp −− ; - X with probability p ; 0 with probability e p . (a) Describe the underlying space S of this random experiment and show the mapping from S to XY S , the range of the pair ( X , Y ). (b) Find the probabilities for all values of ( X , Y ).

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HW5_Spring11_sol - ELEC210 Spring 2011 Homework 5 Solution...

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