(One intermediate revision by the same user not shown) | |||
Line 22: | Line 22: | ||
---- | ---- | ||
==Question== | ==Question== | ||
− | ''' | + | '''1''' |
+ | |||
+ | Let <math class="inline">\mathbf{U}_{n}</math> be a sequence of independent, identically distributed zero-mean, unit-variance Gaussian random variables. The sequence <math class="inline">\mathbf{X}_{n}</math> , <math class="inline">n\geq1</math> , is given by <math class="inline">\mathbf{X}_{n}=\frac{1}{2}\mathbf{U}_{n}+\left(\frac{1}{2}\right)^{2}\mathbf{U}_{n-1}+\cdots+\left(\frac{1}{2}\right)^{n}\mathbf{U}_{1}.</math> | ||
+ | |||
+ | '''(a) (15 points)''' | ||
+ | |||
+ | Find the mean and variance of <math class="inline">\mathbf{X}_{n}</math> . | ||
+ | |||
+ | '''(b) (15 points)''' | ||
+ | |||
+ | Find the characteristic function of <math class="inline">\mathbf{X}_{n}</math> . | ||
+ | |||
+ | '''(c) (10 points)''' | ||
+ | |||
+ | Does the sequence <math class="inline">\mathbf{X}_{n}</math> converge in distribution? A simple yes or no answer is not sufficient. You must justify your answer. | ||
− | |||
:'''Click [[ECE_PhD_QE_CNSIP_2006_Problem1.1|here]] to view student [[ECE_PhD_QE_CNSIP_2006_Problem1.1|answers and discussions]]''' | :'''Click [[ECE_PhD_QE_CNSIP_2006_Problem1.1|here]] to view student [[ECE_PhD_QE_CNSIP_2006_Problem1.1|answers and discussions]]''' | ||
---- | ---- | ||
− | ''' | + | '''2''' |
+ | |||
+ | Let <math class="inline">\Phi</math> be the standard normal distribution, i.e., the distribution function of a zero-mean, unit-variance Gaussian random variable. Let <math class="inline">\mathbf{X}</math> be a normal random variable with mean <math class="inline">\mu</math> and variance 1 . We want to find <math class="inline">E\left[\Phi\left(\mathbf{X}\right)\right]</math> . | ||
+ | |||
+ | '''(a) (10 points)''' | ||
+ | |||
+ | First show that <math class="inline">E\left[\Phi\left(\mathbf{X}\right)\right]=P\left(\mathbf{Z}\leq\mathbf{X}\right)</math> , where <math class="inline">\mathbf{Z}</math> is a standard normal random variable independent of <math class="inline">\mathbf{X}</math> . Hint: Use an intermediate random variable <math class="inline">\mathbf{I}</math> defined as | ||
+ | |||
+ | |||
+ | '''(b) (10 points)''' | ||
+ | |||
+ | Now use the result from Part (a) to show that <math class="inline">E\left[\Phi\left(\mathbf{X}\right)\right]=\Phi\left(\frac{\mu}{\sqrt{2}}\right)</math> . | ||
− | |||
:'''Click [[ECE_PhD_QE_CNSIP_2006_Problem1.2|here]] to view student [[ECE_PhD_QE_CNSIP_2006_Problem1.2|answers and discussions]]''' | :'''Click [[ECE_PhD_QE_CNSIP_2006_Problem1.2|here]] to view student [[ECE_PhD_QE_CNSIP_2006_Problem1.2|answers and discussions]]''' | ||
---- | ---- | ||
− | ''' | + | '''3 (15 points)''' |
+ | |||
+ | Let <math class="inline">\mathbf{Y}(t)</math> be the output of linear system with impulse response <math class="inline">h\left(t\right)</math> and input <math class="inline">\mathbf{X}\left(t\right)+\mathbf{N}\left(t\right)</math> , where <math class="inline">\mathbf{X}\left(t\right)</math> and <math class="inline">\mathbf{N}\left(t\right)</math> are jointly wide-sense stationary independent random processes. If <math class="inline">\mathbf{Z}\left(t\right)=\mathbf{X}\left(t\right)-\mathbf{Y}\left(t\right)</math> , find the power spectral density <math class="inline">S_{\mathbf{Z}}\left(\omega\right)</math> in terms of <math class="inline">S_{\mathbf{X}}\left(\omega\right) , S_{\mathbf{N}}\left(\omega\right) , m_{\mathbf{X}}=E\left[\mathbf{X}\right]</math> , and <math class="inline">m_{\mathbf{Y}}=E\left[\mathbf{Y}\right]</math> . | ||
− | |||
:'''Click [[ECE_PhD_QE_CNSIP_2006_Problem1.3|here]] to view student [[ECE_PhD_QE_CNSIP_2006_Problem1.3|answers and discussions]]''' | :'''Click [[ECE_PhD_QE_CNSIP_2006_Problem1.3|here]] to view student [[ECE_PhD_QE_CNSIP_2006_Problem1.3|answers and discussions]]''' | ||
---- | ---- | ||
− | ''' | + | '''4''' |
+ | |||
+ | Suppose customer orders arrive according to an i.i.d. Bernoulli random process <math class="inline">\mathbf{X}_{n}</math> with parameter <math class="inline">p</math> . Thus, an order arrives at time index <math class="inline">n</math> (i.e., <math class="inline">\mathbf{X}_{n}=1</math> ) with probability <math class="inline">p</math> ; if an order does not arrive at time index <math class="inline">n</math> , then <math class="inline">\mathbf{X}_{n}=0</math> . When an order arrives, its size is an exponential random variable with parameter <math class="inline">\lambda</math> . Let <math class="inline">\mathbf{S}_{n}</math> be the total size of all orders up to time <math class="inline">n</math> . | ||
+ | |||
+ | '''(a) (20 points)''' | ||
+ | |||
+ | Find the mean and autocorrelation function of <math class="inline">\mathbf{S}_{n}</math> . | ||
+ | |||
+ | '''(b) (5 points)''' | ||
+ | |||
+ | Is <math class="inline">\mathbf{S}_{n}</math> a stationary random process? Explain. | ||
− | |||
:'''Click [[ECE_PhD_QE_CNSIP_2006_Problem1.4|here]] to view student [[ECE_PhD_QE_CNSIP_2006_Problem1.4|answers and discussions]]''' | :'''Click [[ECE_PhD_QE_CNSIP_2006_Problem1.4|here]] to view student [[ECE_PhD_QE_CNSIP_2006_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]] |
Latest revision as of 10:35, 10 March 2015
Communication, Networking, Signal and Image Processing (CS)
Question 1: Probability and Random Processes
August 2006
Question
1
Let $ \mathbf{U}_{n} $ be a sequence of independent, identically distributed zero-mean, unit-variance Gaussian random variables. The sequence $ \mathbf{X}_{n} $ , $ n\geq1 $ , is given by $ \mathbf{X}_{n}=\frac{1}{2}\mathbf{U}_{n}+\left(\frac{1}{2}\right)^{2}\mathbf{U}_{n-1}+\cdots+\left(\frac{1}{2}\right)^{n}\mathbf{U}_{1}. $
(a) (15 points)
Find the mean and variance of $ \mathbf{X}_{n} $ .
(b) (15 points)
Find the characteristic function of $ \mathbf{X}_{n} $ .
(c) (10 points)
Does the sequence $ \mathbf{X}_{n} $ converge in distribution? A simple yes or no answer is not sufficient. You must justify your answer.
- Click here to view student answers and discussions
2
Let $ \Phi $ be the standard normal distribution, i.e., the distribution function of a zero-mean, unit-variance Gaussian random variable. Let $ \mathbf{X} $ be a normal random variable with mean $ \mu $ and variance 1 . We want to find $ E\left[\Phi\left(\mathbf{X}\right)\right] $ .
(a) (10 points)
First show that $ E\left[\Phi\left(\mathbf{X}\right)\right]=P\left(\mathbf{Z}\leq\mathbf{X}\right) $ , where $ \mathbf{Z} $ is a standard normal random variable independent of $ \mathbf{X} $ . Hint: Use an intermediate random variable $ \mathbf{I} $ defined as
(b) (10 points)
Now use the result from Part (a) to show that $ E\left[\Phi\left(\mathbf{X}\right)\right]=\Phi\left(\frac{\mu}{\sqrt{2}}\right) $ .
- Click here to view student answers and discussions
3 (15 points)
Let $ \mathbf{Y}(t) $ be the output of linear system with impulse response $ h\left(t\right) $ and input $ \mathbf{X}\left(t\right)+\mathbf{N}\left(t\right) $ , where $ \mathbf{X}\left(t\right) $ and $ \mathbf{N}\left(t\right) $ are jointly wide-sense stationary independent random processes. If $ \mathbf{Z}\left(t\right)=\mathbf{X}\left(t\right)-\mathbf{Y}\left(t\right) $ , find the power spectral density $ S_{\mathbf{Z}}\left(\omega\right) $ in terms of $ S_{\mathbf{X}}\left(\omega\right) , S_{\mathbf{N}}\left(\omega\right) , m_{\mathbf{X}}=E\left[\mathbf{X}\right] $ , and $ m_{\mathbf{Y}}=E\left[\mathbf{Y}\right] $ .
- Click here to view student answers and discussions
4
Suppose customer orders arrive according to an i.i.d. Bernoulli random process $ \mathbf{X}_{n} $ with parameter $ p $ . Thus, an order arrives at time index $ n $ (i.e., $ \mathbf{X}_{n}=1 $ ) with probability $ p $ ; if an order does not arrive at time index $ n $ , then $ \mathbf{X}_{n}=0 $ . When an order arrives, its size is an exponential random variable with parameter $ \lambda $ . Let $ \mathbf{S}_{n} $ be the total size of all orders up to time $ n $ .
(a) (20 points)
Find the mean and autocorrelation function of $ \mathbf{S}_{n} $ .
(b) (5 points)
Is $ \mathbf{S}_{n} $ a stationary random process? Explain.
- Click here to view student answers and discussions