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Unformatted text preview: ELEC210 Spring 2011 Homework4Solution
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 piecewise 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 zeromean 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).^(11).*exp(1.*1.*x)./gamma(1);plot(x,y,'r');hold on;
y=1.*(1.*x).^(21).*exp(1.*1.*x)./gamma(2);plot(x,y,'g')
y=1.*(1.*x).^(31).*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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 Spring '11
 Prof.ShenghuiSong

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