View source for ECE-QE CS1-2012 - Rhea

[[Category:ECE]]
[[Category:QE]]
[[Category:CNSIP]]
[[Category:problem solving]]
[[Category:random variables]]
[[Category:probability]]

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[[ECE_PhD_Qualifying_Exams|ECE Ph.D. Qualifying Exam]]
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Communication, Networking, Signal and Image Processing (CS)

Question 1: Probability and Random Processes
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August 2012
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==Question==
'''Part 1. ''' 25 pts


State and prove the Chebyshev inequality for random variable <math class="inline">\mathbf{X}</math>  with mean <math class="inline">\mu</math> and variance <math class="inline">\sigma^2</math>. In constructing your proof, keep in mind that <math class="inline">\mathbf{X}</math> may be either a discrete or continuous random variable.



:'''Click [[ECE-QE_CS1-2012_solusion-1|here]] to view student [[ECE-QE_CS1-2012_solusion-1|answers and discussions]]'''
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'''Part 2.''' 25 pts


Let <math class="inline">\mathbf{X}_{1} \dots \mathbf{X}_{n} \dots </math> be a sequence of independent, identical distributed random variables, each uniformly distributed on the interval [0, 1], an hence having pdf 
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<math class="inline">f_{X}\left(x\right)=\begin{cases}
\begin{array}{lll}
1,    \text{      for }  0 \leq x \leq1\\
0,   \text{      elsewhere. }
\end{array}\end{cases}</math>
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Let <math class="inline">\mathbf{Y}_{n}</math> be a new random variable defined by
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 <math class="inline">\mathbf{Y}_{n} = min \,\{{ \mathbf{X}_1, \mathbf{X}_2, \dots \mathbf{X}_n} \}</math>
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'''(a)''' Find the pdf of <math class="inline">\mathbf{Y}_{n}</math>

'''(b)''' Does the sequence <math class="inline">\mathbf{Y}_{n}</math> converge in probability?

'''(c)''' Does the sequence <math class="inline">\mathbf{Y}_{n}</math> converge in distribution? If yes, specify the cumulative function of the random variable it converges to.


:'''Click [[ECE-QE_CS1-2012_solusion-2|here]] to view student [[ECE-QE_CS1-2012_solusion-2|answers and discussions]]'''
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