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= [[ECE_PhD_Qualifying_Exams|ECE Ph.D. Qualifying Exam]]: COMMUNICATIONS, NETWORKING, SIGNAL AND IMAGE PROESSING (CS)- Question 1, January 2004=
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[[ECE_PhD_Qualifying_Exams|ECE Ph.D. Qualifying Exam]]
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Communication, Networking, Signal and Image Processing (CS)
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Question 1: Probability and Random Processes
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January 2004
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==Question==
 
==Question==
'''Part 1. '''
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'''1. (30 pts.)'''
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This question consists of two separate short questions relating to the structure of probability space:
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'''(a)'''
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Assume that <math class="inline">\mathcal{S}</math>  is the sample space of a random experiment and that <math class="inline">\mathcal{F}_{1}</math>  and <math class="inline">\mathcal{F}_{2}</math>  are <math class="inline">\sigma</math> -fields (valid event spaces) on <math class="inline">\mathcal{S}</math> . Prove that <math class="inline">\mathcal{F}_{1}\cap\mathcal{F}_{2}</math>  is also a <math class="inline">\sigma</math> -field on <math class="inline">S</math> .
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'''(b)'''
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Consider a sample space <math class="inline">\mathcal{S}</math>  and corresponding event space <math class="inline">\mathcal{F}</math> . Suppose that <math class="inline">P_{1}</math>  and <math class="inline">P_{2}</math>  are both balid probability measures defined on <math class="inline">\mathcal{F}</math> . Prove that <math class="inline">P</math>  defined by <math class="inline">P\left(A\right)=\alpha_{1}P_{1}\left(A\right)+\alpha_{2}P_{2}\left(A\right),\qquad\forall A\in\mathcal{F}</math>  is also a valid probability measure on <math class="inline">\mathcal{F}</math>  if <math class="inline">\alpha_{1},\;\alpha_{2}\geq0</math>  and <math class="inline">\alpha_{1}+\alpha_{2}=1</math> .
  
Write Statement here
 
  
 
:'''Click [[ECE_PhD_QE_CNSIP_Jan_2004_Problem1.1|here]] to view student [[ECE_PhD_QE_CNSIP_Jan_2004_Problem1.1|answers and discussions]]'''
 
:'''Click [[ECE_PhD_QE_CNSIP_Jan_2004_Problem1.1|here]] to view student [[ECE_PhD_QE_CNSIP_Jan_2004_Problem1.1|answers and discussions]]'''
 
----
 
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'''Part 2.'''
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'''2. (10 pts.)'''
  
Write question here.
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Identical twins come from the same egg and and hence are of the same sex. Fraternal twins have a probability <math class="inline">1/2</math>  of being of the same sex. Among twins, the probability of a fraternal set is p  and of an identical set is <math class="inline">q=1-p</math> . Given that a set of twins selected at random are of the same sex, what is the probability they are fraternal? (Simplify your answer as much as possible.) Sketch a plot of the conditional probability that the twins are fraternal given that they are of the same sex as a function of <math class="inline">q</math>  (the probability that a set of twins are identical.)
  
 
:'''Click [[ECE_PhD_QE_CNSIP_Jan_2004_Problem1.2|here]] to view student [[ECE_PhD_QE_CNSIP_Jan_2004_Problem1.2|answers and discussions]]'''
 
:'''Click [[ECE_PhD_QE_CNSIP_Jan_2004_Problem1.2|here]] to view student [[ECE_PhD_QE_CNSIP_Jan_2004_Problem1.2|answers and discussions]]'''
 
----
 
----
'''Part 3.'''
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'''3. (30 pts.)'''
  
Write question here.
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Let <math class="inline">\mathbf{X}\left(t\right)</math>  be a real continuous-time Gaussian random process. Show that its probabilistic behavior is completely characterized by its mean <math class="inline">\mu_{\mathbf{X}}\left(t\right)=E\left[\mathbf{X}\left(t\right)\right]</math> and its autocorrelation function <math class="inline">R_{\mathbf{XX}}\left(t_{1},t_{2}\right)=E\left[\mathbf{X}\left(t_{1}\right)\mathbf{X}\left(t_{2}\right)\right].</math>
  
 
:'''Click [[ECE_PhD_QE_CNSIP_Jan_2004_Problem1.3|here]] to view student [[ECE_PhD_QE_CNSIP_Jan_2004_Problem1.3|answers and discussions]]'''
 
:'''Click [[ECE_PhD_QE_CNSIP_Jan_2004_Problem1.3|here]] to view student [[ECE_PhD_QE_CNSIP_Jan_2004_Problem1.3|answers and discussions]]'''
 
----
 
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'''Part 4.'''
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'''4. (30 pts.)'''
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Assume that <math class="inline">\mathbf{X}\left(t\right)</math>  is a zero-mean, continuous-time, Gaussian white noise process with autocorrelation function <math class="inline">R_{\mathbf{XX}}\left(t_{1},t_{2}\right)=\delta\left(t_{1}-t_{2}\right)</math>.  Let <math class="inline">\mathbf{Y}\left(t\right)</math>  be a new random process defined by <math class="inline">\mathbf{Y}\left(t\right)=\frac{1}{T}\int_{t-T}^{t}\mathbf{X}\left(s\right)ds</math>,  where <math class="inline">T>0</math> .
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'''(a)'''
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What is the mean of <math class="inline">\mathbf{Y}\left(t\right)</math> ?
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'''(b)'''
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What is the autocorrelation function of <math class="inline">\mathbf{Y}\left(t\right)</math> ?
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'''(c)'''
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Write an expression for the second-order pdf <math class="inline">f_{\mathbf{Y}\left(t_{1}\right)\mathbf{Y}\left(t_{2}\right)}\left(y_{1},y_{2}\right)</math>  of <math class="inline">\mathbf{Y}\left(t\right)</math> .
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 +
(d)
  
Write question here.
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Under what conditions on <math class="inline">t_{1}</math>  and <math class="inline">t_{2}</math>  will <math class="inline">\mathbf{Y}\left(t_{1}\right)</math>  and <math class="inline">\mathbf{Y}\left(t_{2}\right)</math>  be statistically independent?
  
 
:'''Click [[ECE_PhD_QE_CNSIP_Jan_2004_Problem1.4|here]] to view student [[ECE_PhD_QE_CNSIP_Jan_2004_Problem1.4|answers and discussions]]'''
 
:'''Click [[ECE_PhD_QE_CNSIP_Jan_2004_Problem1.4|here]] to view student [[ECE_PhD_QE_CNSIP_Jan_2004_Problem1.4|answers and discussions]]'''
 
----
 
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[[ECE_PhD_Qualifying_Exams|Back to ECE Qualifying Exams (QE) page]]
 
[[ECE_PhD_Qualifying_Exams|Back to ECE Qualifying Exams (QE) page]]

Latest revision as of 01:01, 10 March 2015


ECE Ph.D. Qualifying Exam

Communication, Networking, Signal and Image Processing (CS)

Question 1: Probability and Random Processes

January 2004



Question

1. (30 pts.)

This question consists of two separate short questions relating to the structure of probability space:

(a)

Assume that $ \mathcal{S} $ is the sample space of a random experiment and that $ \mathcal{F}_{1} $ and $ \mathcal{F}_{2} $ are $ \sigma $ -fields (valid event spaces) on $ \mathcal{S} $ . Prove that $ \mathcal{F}_{1}\cap\mathcal{F}_{2} $ is also a $ \sigma $ -field on $ S $ .

(b)

Consider a sample space $ \mathcal{S} $ and corresponding event space $ \mathcal{F} $ . Suppose that $ P_{1} $ and $ P_{2} $ are both balid probability measures defined on $ \mathcal{F} $ . Prove that $ P $ defined by $ P\left(A\right)=\alpha_{1}P_{1}\left(A\right)+\alpha_{2}P_{2}\left(A\right),\qquad\forall A\in\mathcal{F} $ is also a valid probability measure on $ \mathcal{F} $ if $ \alpha_{1},\;\alpha_{2}\geq0 $ and $ \alpha_{1}+\alpha_{2}=1 $ .


Click here to view student answers and discussions

2. (10 pts.)

Identical twins come from the same egg and and hence are of the same sex. Fraternal twins have a probability $ 1/2 $ of being of the same sex. Among twins, the probability of a fraternal set is p and of an identical set is $ q=1-p $ . Given that a set of twins selected at random are of the same sex, what is the probability they are fraternal? (Simplify your answer as much as possible.) Sketch a plot of the conditional probability that the twins are fraternal given that they are of the same sex as a function of $ q $ (the probability that a set of twins are identical.)

Click here to view student answers and discussions

3. (30 pts.)

Let $ \mathbf{X}\left(t\right) $ be a real continuous-time Gaussian random process. Show that its probabilistic behavior is completely characterized by its mean $ \mu_{\mathbf{X}}\left(t\right)=E\left[\mathbf{X}\left(t\right)\right] $ and its autocorrelation function $ R_{\mathbf{XX}}\left(t_{1},t_{2}\right)=E\left[\mathbf{X}\left(t_{1}\right)\mathbf{X}\left(t_{2}\right)\right]. $

Click here to view student answers and discussions

4. (30 pts.)

Assume that $ \mathbf{X}\left(t\right) $ is a zero-mean, continuous-time, Gaussian white noise process with autocorrelation function $ R_{\mathbf{XX}}\left(t_{1},t_{2}\right)=\delta\left(t_{1}-t_{2}\right) $. Let $ \mathbf{Y}\left(t\right) $ be a new random process defined by $ \mathbf{Y}\left(t\right)=\frac{1}{T}\int_{t-T}^{t}\mathbf{X}\left(s\right)ds $, where $ T>0 $ .

(a)

What is the mean of $ \mathbf{Y}\left(t\right) $ ?

(b)

What is the autocorrelation function of $ \mathbf{Y}\left(t\right) $ ?

(c)

Write an expression for the second-order pdf $ f_{\mathbf{Y}\left(t_{1}\right)\mathbf{Y}\left(t_{2}\right)}\left(y_{1},y_{2}\right) $ of $ \mathbf{Y}\left(t\right) $ .

(d)

Under what conditions on $ t_{1} $ and $ t_{2} $ will $ \mathbf{Y}\left(t_{1}\right) $ and $ \mathbf{Y}\left(t_{2}\right) $ be statistically independent?

Click here to view student answers and discussions

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