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| − | State and prove the Chebyshev inequality for random variable <math | + | State and prove the Chebyshev inequality for random variable <math>\mathbf{X}</math> with mean <math>\mu</math> and variance <math>\sigma^2</math>. In constructing your proof, keep in mind that <math>\mathbf{X}</math> may be either a discrete or continuous random variable. |
Revision as of 08:52, 26 January 2014
Communication, Networking, Signal and Image Processing (CS)
Question 1: Probability and Random Processes
August 2012
Question
Problem 2. 25 pts
State and prove the Chebyshev inequality for random variable $ \mathbf{X} $ with mean $ \mu $ and variance $ \sigma^2 $. In constructing your proof, keep in mind that $ \mathbf{X} $ may be either a discrete or continuous random variable.
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Problem 3. 25 pts
Let $ \mathbf{X}_{1} \dots \mathbf{X}_{n} \dots $ be a sequence of independent, identical distributed random variables, each uniformly distributed on the interval [0, 1], an hence having pdf
$ f_{X}\left(x\right)=\begin{cases} \begin{array}{lll} 1, \text{ for } 0 \leq x \leq1\\ 0, \text{ elsewhere. } \end{array}\end{cases} $
Let $ \mathbf{Y}_{n} $ be a new random variable defined by
$ \mathbf{Y}_{n} = min \,\{{ \mathbf{X}_1, \mathbf{X}_2, \dots \mathbf{X}_n} \} $
(a) Find the pdf of $ \mathbf{Y}_{n} $.
(b) Does the sequence $ \mathbf{Y}_{n} $ converge in probability?
(c) Does the sequence $ \mathbf{Y}_{n} $ converge in distribution? If yes, specify the cumulative function of the random variable it converges to.
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