HW4_Spring11_Sol

HW4_Spring11_Sol - ELEC210 Spring 2011 Homework-4-Solution...

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Unformatted text preview: ELEC210 Spring 2011 Homework-4-Solution 1. The random variable X has cdf as shown in the figure below: (a) What type of random variable is X? (b) Find the following probabilities: , , , , , . Solution(16’): (a) (4’)The cdf is neither continuous nor piece-wise constant, so it’s mixed type. (b) (12’) , , , , , . 2. A point is selected at random inside a square defined by . Assume the point is equally likely to fall anywhere in the square. Let the random variable Z be given by the minimum of the two coordinates of the point. (a) Find the sample space S of the coordinates of the point and the sample space of Z, (b) Show the mapping from S to . (c) Find the region in the square corresponding to the event . (d) Find the cdf of Z. (e) Use the cdf to find: . . Solution(16’): (a) (3’) , . (b) (3’)The two segments as shown by the bold black curve in S correspond to point a in b z 0 z (c) (3’)The region corresponding to the event (d) (3’) a 0 b is shown by the blue area. . (e) (4’) , , , . 3. A random variable X has pdf: (a) Find c. (b) Find . (c) Find . (d) Find the mean and variance of X. Solution(16’): (a) (4’) so (b) (4’) , . . b (c) (4’) . (d) (4’) , , so . 4. A limiter is shown in figure below: (a) Find an expression for the mean and variance of for an arbitrary continuous random variable X. (b) Evaluate the mean and variance if X is a Laplacian random variable with . g(x) a a -a x -a Solution(16’): (a) (8’) , , . (b) (8’)If X is Laplacian with , then . , so . 5. A binary transmission system transmits a signal X ( -1 to send a “0” bit; +1 to send a “1” bit). The received signal is Y=X+N where noise N has a zero-mean Gaussian distribution with variance . Assume that “0” bits are three times likely as “1” bits. (a) Find the conditional pdf of Y given the input value: and . (b) The receiver decides a “0” was transmitted if the observed value of y satisfies and it decides a “1” was transmitted otherwise. What is the probability that the receiver makes an error given that a “1” was transmitted? A “0” was transmitted? Assume . (c) What is the overall probability of error? Solution(18’): (a) (6’) , . (b) (8’)Since “0” bits are three times likely as “1” bits, . If a “1” was transmitted, the receiver will make an error when it decides “0”, that is y satisfies , so If a “0” was transmitted, the receiver will make an error when it decides “1”, that is y does not satisfy , which means , (c) (4’) . 6. (a) Plot the pdf of a Gamma random variable for and λ with Matlab (Put the three curves in one figure). (b) Plot the pdf of a Laplacian random variable for with Matlab (Put the three curves in one figure). Solution(18’): (a) (9’)The Matlab Code: x=0:0.01:5; y=1.*(1.*x).^(1-1).*exp(-1.*1.*x)./gamma(1);plot(x,y,'r');hold on; y=1.*(1.*x).^(2-1).*exp(-1.*1.*x)./gamma(2);plot(x,y,'g') y=1.*(1.*x).^(3-1).*exp(-1.*1.*x)./gamma(3);plot(x,y,'b') The figure is: (b) (9’)The Matlab Code: x=-5:0.01:5; y=1/2.*exp(-1.*abs(x));plot(x,y,'r');hold on; y=2/2.*exp(-2.*abs(x));plot(x,y,'g') y=3/2.*exp(-3.*abs(x));plot(x,y,'b') The figure is: ...
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