(New page: Category:ECE Category:QE Category:CNSIP Category:problem solving Category:random variables Category:probability <center> <font size= 4> [[ECE_PhD_Qualifying_Exams|...) |
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'''Part 1. ''' | '''Part 1. ''' | ||
| − | + | Consider <math class="inline">n</math> independent flips of a coin having probability <math class="inline">p</math> of landing on heads. Say that a changeover occurs whenever an outcome differs from the one preceding it. For instance, if <math class="inline">n=5</math> and the sequence <math class="inline">HHTHT</math> is observed, then there are 3 changeovers. Find the expected number of changeovers for <math class="inline">n</math> flips. ''Hint'': Express the number of changeovers as a sum of Bernoulli random variables. | |
:'''Click [[ECE_PhD_QE_CNSIP_2013_Problem1.1|here]] to view student [[ECE_PhD_QE_CNSIP_2013_Problem1.1|answers and discussions]]''' | :'''Click [[ECE_PhD_QE_CNSIP_2013_Problem1.1|here]] to view student [[ECE_PhD_QE_CNSIP_2013_Problem1.1|answers and discussions]]''' | ||
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'''Part 2.''' | '''Part 2.''' | ||
| − | + | Let <math>X_1,X_2,...</math> be a sequence of jointly Gaussian random variables with covariance | |
| + | |||
| + | <math>Cov(X_i,X_j) = \left\{ \begin{array}{ll} | ||
| + | {\sigma}^2, & i=j\\ | ||
| + | \rho{\sigma}^2, & |i-j|=1\\ | ||
| + | 0, & otherwise | ||
| + | \end{array} \right.</math> | ||
| + | |||
| + | Suppose we take 2 consecutive samples from this sequence to form a vector <math>X</math>, which is then linearly transformed to form a 2-dimensional random vector <math>Y=AX</math>. Find a matrix <math>A</math> so that the components of <math>Y</math> are independent random variables You must justify your answer. | ||
:'''Click [[ECE_PhD_QE_CNSIP_2013_Problem1.2|here]] to view student [[ECE_PhD_QE_CNSIP_2013_Problem1.2|answers and discussions]]''' | :'''Click [[ECE_PhD_QE_CNSIP_2013_Problem1.2|here]] to view student [[ECE_PhD_QE_CNSIP_2013_Problem1.2|answers and discussions]]''' | ||
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'''Part 3.''' | '''Part 3.''' | ||
| − | + | Let <math>X</math> be an exponential random variable with parameter <math>\lambda</math>, so that <math>f_X(x)=\lambda{exp}(-\lambda{x})u(x)</math>. Find the variance of <math>X</math>. You must show all of your work. | |
:'''Click [[ECE_PhD_QE_CNSIP_2013_Problem1.3|here]] to view student [[ECE_PhD_QE_CNSIP_2013_Problem1.3|answers and discussions]]''' | :'''Click [[ECE_PhD_QE_CNSIP_2013_Problem1.3|here]] to view student [[ECE_PhD_QE_CNSIP_2013_Problem1.3|answers and discussions]]''' | ||
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'''Part 4.''' | '''Part 4.''' | ||
| − | + | Consider a sequence of independent random variables <math>X_1,X_2,...</math>, where <math>X_n</math> has pdf | |
| + | |||
| + | <math>\begin{align}f_n(x)=&(1-\frac{1}{n})\frac{1}{\sqrt{2\pi}\sigma}exp[-\frac{1}{2\sigma^2}(x-\frac{n-1}{n}\sigma)^2]\\ | ||
| + | &+\frac{1}{n}\sigma exp(-\sigma x)u(x)\end{align}</math>. | ||
| + | |||
| + | Does this sequence converge in the mean-square sense? ''Hint:'' Use the Cauchy criterion for mean-square convergence, which states that a sequence of random variables <math>X_1,X_2,...</math> converges in mean-square if and only if <math>E[|X_n-X_{n+m}|] \to 0</math> as <math>n \to \infty</math>, for every <math>m>0</math>. | ||
:'''Click [[ECE_PhD_QE_CNSIP_2013_Problem1.4|here]] to view student [[ECE_PhD_QE_CNSIP_2013_Problem1.4|answers and discussions]]''' | :'''Click [[ECE_PhD_QE_CNSIP_2013_Problem1.4|here]] to view student [[ECE_PhD_QE_CNSIP_2013_Problem1.4|answers and discussions]]''' | ||
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Latest revision as of 06:47, 4 November 2014
Communication, Networking, Signal and Image Processing (CS)
Question 1: Probability and Random Processes
August 2013
Question
Part 1.
Consider $ n $ independent flips of a coin having probability $ p $ of landing on heads. Say that a changeover occurs whenever an outcome differs from the one preceding it. For instance, if $ n=5 $ and the sequence $ HHTHT $ is observed, then there are 3 changeovers. Find the expected number of changeovers for $ n $ flips. Hint: Express the number of changeovers as a sum of Bernoulli random variables.
- Click here to view student answers and discussions
Part 2.
Let $ X_1,X_2,... $ be a sequence of jointly Gaussian random variables with covariance
$ Cov(X_i,X_j) = \left\{ \begin{array}{ll} {\sigma}^2, & i=j\\ \rho{\sigma}^2, & |i-j|=1\\ 0, & otherwise \end{array} \right. $
Suppose we take 2 consecutive samples from this sequence to form a vector $ X $, which is then linearly transformed to form a 2-dimensional random vector $ Y=AX $. Find a matrix $ A $ so that the components of $ Y $ are independent random variables You must justify your answer.
- Click here to view student answers and discussions
Part 3.
Let $ X $ be an exponential random variable with parameter $ \lambda $, so that $ f_X(x)=\lambda{exp}(-\lambda{x})u(x) $. Find the variance of $ X $. You must show all of your work.
- Click here to view student answers and discussions
Part 4.
Consider a sequence of independent random variables $ X_1,X_2,... $, where $ X_n $ has pdf
$ \begin{align}f_n(x)=&(1-\frac{1}{n})\frac{1}{\sqrt{2\pi}\sigma}exp[-\frac{1}{2\sigma^2}(x-\frac{n-1}{n}\sigma)^2]\\ &+\frac{1}{n}\sigma exp(-\sigma x)u(x)\end{align} $.
Does this sequence converge in the mean-square sense? Hint: Use the Cauchy criterion for mean-square convergence, which states that a sequence of random variables $ X_1,X_2,... $ converges in mean-square if and only if $ E[|X_n-X_{n+m}|] \to 0 $ as $ n \to \infty $, for every $ m>0 $.
- Click here to view student answers and discussions
