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----==Question== | ----==Question== | ||
| − | ''' | + | '''1''' |
| + | |||
| + | The Weak Law of Large Numbers states that if <math class="inline">\mathbf{X}_{1},\mathbf{X}_{2},\mathbf{X}_{3},\cdots</math> is a sequence of i.i.d. random variables with finite mean <math class="inline">E\left[\mathbf{X}_{i}\right]=\mu</math> for every <math class="inline">i</math> , then the sample mean <math class="inline">\mathbf{Y}_{n}=\frac{1}{n}\sum_{i=1}^{n}\mathbf{X}_{i}</math> converges to <math class="inline">\mu</math> in probability. Suppose that instead of being i.i.d , <math class="inline">\mathbf{X}_{1},\mathbf{X}_{2},\mathbf{X}_{3},\cdots</math> each have finite mean <math class="inline">\mu</math> , and the covariance function of the sequence <math class="inline">\mathbf{X}_{n}</math> is <math class="inline">Cov\left(\mathbf{X}_{i},\mathbf{X}_{j}\right)=\sigma^{2}\rho^{\left|i-j\right|}</math> , where <math class="inline">\left|\rho\right|<1</math> and <math class="inline">\sigma^{2}>0</math> . | ||
| + | |||
| + | a. (13 points) | ||
| + | |||
| + | Find the mean and variance of <math class="inline">\mathbf{Y}_{n}</math> . | ||
| + | |||
| + | b. (12 points) | ||
| + | |||
| + | Does the sample mean still converge to <math class="inline">\mu</math> in probability? You must justify your answer. | ||
| + | |||
| − | |||
:'''Click [[ECE_PhD_QE_CNSIP_2008_Problem1.1|here]] to view student [[ECE_PhD_QE_CNSIP_2008_Problem1.1|answers and discussions]]''' | :'''Click [[ECE_PhD_QE_CNSIP_2008_Problem1.1|here]] to view student [[ECE_PhD_QE_CNSIP_2008_Problem1.1|answers and discussions]]''' | ||
---- | ---- | ||
| − | ''' | + | '''2.''' |
| + | |||
| + | Let <math class="inline">\mathbf{X}_{1},\mathbf{X}_{2},\mathbf{X}_{3},\cdots</math> be a sequence of i.i.d Bernoulli random variables with <math class="inline">p=1/2</math> , and let <math class="inline">\mathbf{Y}_{n}=2^{n}\mathbf{X}_{1}\mathbf{X}_{2}\cdots\mathbf{X}_{n}</math> . | ||
| + | |||
| + | a. (15 points) | ||
| + | |||
| + | Does the sequence <math class="inline">\mathbf{Y}_{n}</math> converge to <math class="inline">0</math> almost everywhere? | ||
| + | |||
| + | b. (15 points) | ||
| + | |||
| + | Does the sequence <math class="inline">\mathbf{Y}_{n}</math> converge to 0 in the mean square sense? | ||
| + | |||
| − | |||
:'''Click [[ECE_PhD_QE_CNSIP_2008_Problem1.2|here]] to view student [[ECE_PhD_QE_CNSIP_2008_Problem1.2|answers and discussions]]''' | :'''Click [[ECE_PhD_QE_CNSIP_2008_Problem1.2|here]] to view student [[ECE_PhD_QE_CNSIP_2008_Problem1.2|answers and discussions]]''' | ||
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| − | ''' | + | '''3.''' |
| + | |||
| + | Consider a random process <math class="inline">\mathbf{X}\left(t\right)</math> that assumes values <math class="inline">\pm1</math> . Suppose that <math class="inline">\mathbf{X}\left(0\right)=\pm1</math> with probability <math class="inline">1/2</math> , and suppose that <math class="inline">\mathbf{X}\left(t\right)</math> then changes polarity with each occurrence of an event in a Poisson process of rate <math class="inline">\lambda</math> . | ||
| + | |||
| + | Note: | ||
| + | |||
| + | You might find the equations <math class="inline">\frac{1}{2}\left(e^{x}+e^{-x}\right)=\sum_{j=0}^{\infty}\frac{x^{2j}}{\left(2j\right)!}</math> and <math class="inline">\frac{1}{2}\left(e^{x}-e^{-x}\right)=\sum_{j=0}^{\infty}\frac{x^{2j+1}}{\left(2j+1\right)!}</math> helpful. | ||
| + | |||
| + | a. (15 points) | ||
| + | |||
| + | Find the probability mass function of <math class="inline">\mathbf{X}\left(t\right)</math> . | ||
| + | |||
| + | b. (15 points) | ||
| + | |||
| + | Find the autocovariance function of the random process <math class="inline">\mathbf{X}\left(t\right)</math> . | ||
| − | |||
:'''Click [[ECE_PhD_QE_CNSIP_2008_Problem1.3|here]] to view student [[ECE_PhD_QE_CNSIP_2008_Problem1.3|answers and discussions]]''' | :'''Click [[ECE_PhD_QE_CNSIP_2008_Problem1.3|here]] to view student [[ECE_PhD_QE_CNSIP_2008_Problem1.3|answers and discussions]]''' | ||
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| − | ''' | + | '''4 (15 points)''' |
| + | |||
| + | Messages arrive at a message center according to a Poisson process of rate <math class="inline">\lambda</math> messages per hour. Every hour the messages that have arrived during the previous hour are forwarded to their destination. Find the expected value of the total time waited by all messages that arrive during the hour. | ||
| − | |||
:'''Click [[ECE_PhD_QE_CNSIP_2008_Problem1.4|here]] to view student [[ECE_PhD_QE_CNSIP_2008_Problem1.4|answers and discussions]]''' | :'''Click [[ECE_PhD_QE_CNSIP_2008_Problem1.4|here]] to view student [[ECE_PhD_QE_CNSIP_2008_Problem1.4|answers and discussions]]''' | ||
| + | ---- | ||
---- | ---- | ||
[[ECE_PhD_Qualifying_Exams|Back to ECE Qualifying Exams (QE) page]] | [[ECE_PhD_Qualifying_Exams|Back to ECE Qualifying Exams (QE) page]] | ||
Revision as of 10:04, 10 March 2015
Communication, Networking, Signal and Image Processing (CS)
Question 1: Probability and Random Processes
August 2008
==Question==
1
The Weak Law of Large Numbers states that if $ \mathbf{X}_{1},\mathbf{X}_{2},\mathbf{X}_{3},\cdots $ is a sequence of i.i.d. random variables with finite mean $ E\left[\mathbf{X}_{i}\right]=\mu $ for every $ i $ , then the sample mean $ \mathbf{Y}_{n}=\frac{1}{n}\sum_{i=1}^{n}\mathbf{X}_{i} $ converges to $ \mu $ in probability. Suppose that instead of being i.i.d , $ \mathbf{X}_{1},\mathbf{X}_{2},\mathbf{X}_{3},\cdots $ each have finite mean $ \mu $ , and the covariance function of the sequence $ \mathbf{X}_{n} $ is $ Cov\left(\mathbf{X}_{i},\mathbf{X}_{j}\right)=\sigma^{2}\rho^{\left|i-j\right|} $ , where $ \left|\rho\right|<1 $ and $ \sigma^{2}>0 $ .
a. (13 points)
Find the mean and variance of $ \mathbf{Y}_{n} $ .
b. (12 points)
Does the sample mean still converge to $ \mu $ in probability? You must justify your answer.
- Click here to view student answers and discussions
2.
Let $ \mathbf{X}_{1},\mathbf{X}_{2},\mathbf{X}_{3},\cdots $ be a sequence of i.i.d Bernoulli random variables with $ p=1/2 $ , and let $ \mathbf{Y}_{n}=2^{n}\mathbf{X}_{1}\mathbf{X}_{2}\cdots\mathbf{X}_{n} $ .
a. (15 points)
Does the sequence $ \mathbf{Y}_{n} $ converge to $ 0 $ almost everywhere?
b. (15 points)
Does the sequence $ \mathbf{Y}_{n} $ converge to 0 in the mean square sense?
- Click here to view student answers and discussions
3.
Consider a random process $ \mathbf{X}\left(t\right) $ that assumes values $ \pm1 $ . Suppose that $ \mathbf{X}\left(0\right)=\pm1 $ with probability $ 1/2 $ , and suppose that $ \mathbf{X}\left(t\right) $ then changes polarity with each occurrence of an event in a Poisson process of rate $ \lambda $ .
Note:
You might find the equations $ \frac{1}{2}\left(e^{x}+e^{-x}\right)=\sum_{j=0}^{\infty}\frac{x^{2j}}{\left(2j\right)!} $ and $ \frac{1}{2}\left(e^{x}-e^{-x}\right)=\sum_{j=0}^{\infty}\frac{x^{2j+1}}{\left(2j+1\right)!} $ helpful.
a. (15 points)
Find the probability mass function of $ \mathbf{X}\left(t\right) $ .
b. (15 points)
Find the autocovariance function of the random process $ \mathbf{X}\left(t\right) $ .
- Click here to view student answers and discussions
4 (15 points)
Messages arrive at a message center according to a Poisson process of rate $ \lambda $ messages per hour. Every hour the messages that have arrived during the previous hour are forwarded to their destination. Find the expected value of the total time waited by all messages that arrive during the hour.
- Click here to view student answers and discussions
