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| − | + | 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. | |
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| − | + | 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 | |
| − | </math></ | + | <br> |
| + | <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> | ||
| + | <br> | ||
| + | |||
| + | Let <math class="inline">\mathbf{Y}_{n}</math> be a new random variable defined by | ||
| + | <br> | ||
| + | <math class="inline">\mathbf{Y}_{n} = min \,\{{ \mathbf{X}_1, \mathbf{X}_2, \dots \mathbf{X}_n} \}</math> | ||
| + | <br> | ||
| + | |||
| + | '''(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. | ||
Revision as of 15:41, 25 January 2014
Communication, Networking, Signal and Image Processing (CS)
Question 1: Probability and Random Processes
August 2012
Question
Part 1. 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.
- Click here to view student answers and discussions
Part 2. 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.
- Click here to view student answers and discussions
