Communication, Networking, Signal and Image Processing (CS)
Question 1: Probability and Random Processes
August 2004
Question
1. (20 pts.)
A probability space $ \left(\mathcal{S},\mathcal{F},\mathcal{P}\right) $ has a sample space consisting of all pairs of positive integers: $ \mathcal{S}=\left\{ \left(k,m\right):\; k=1,2,\cdots;\; m=1,2,\cdots\right\} $ . The event space $ \mathcal{F} $ is the power set of $ \mathcal{S} $ , and the probability measure $ \mathcal{P} $ is specified by the pmf $ p\left(k,m\right)=p^{2}\left(1-p\right)^{k+m-2},\qquad p\in\left(0,1\right) $.
(a)
Find $ P\left(\left\{ \left(k,m\right):\; k\geq m\right\} \right) $ .
(b)
Find $ P\left(\left\{ \left(k,m\right):\; k+m=r\right\} \right) $ , for $ r=2,3,\cdots $ .
(c)
Find $ P\left(\left\{ \left(k,m\right):\; k\text{ is an odd number}\right\} \right) $ .
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2. (20 pts.)
Let $ \mathbf{X} $ and $ \mathbf{Y} $ be two independent identically distributed exponential random variables having mean $ \mu $ . Let $ \mathbf{Z}=\mathbf{X}+\mathbf{Y} $ . Find $ f_{\mathbf{X}}\left(x|\mathbf{Z}=z\right) $ , the conditional pdf of $ \mathbf{X} $ given the event $ \left\{ \mathbf{Z}=z\right\} $ .
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3. (25 pts.)
Let $ \mathbf{X}_{1},\cdots,\mathbf{X}_{n} $ be independent identically distributed (i.i.d. ) random variables uniformaly distributed over the interval $ \left[0,1\right] $ .
(a)
Find the probability density function of $ \mathbf{Y}=\max\left\{ \mathbf{X}_{1},\cdots,\mathbf{X}_{n}\right\} $ .
(b)
Find the probability density function of $ \mathbf{Z}=\min\left\{ \mathbf{X}_{1},\cdots,\mathbf{X}_{n}\right\} $ .
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4. (35 pts.)
Assume that $ \mathbf{X}\left(t\right) $ is a zero-mean, continuous-time, Gaussian white noise process with autocorrelation function $ R_{\mathbf{XX}}\left(t_{1},t_{2}\right)=\frac{N_{0}}{2}\delta\left(t_{1}-t_{2}\right). $ Let $ \mathbf{Y}\left(t\right) $ be a new random process defined as the output of a linear time-invariant system with impulse response $ h\left(t\right)=\frac{1}{T}e^{-t/T}\cdot u\left(t\right), $ where $ u\left(t\right) $ is the unit step function and $ T>0 $ .
(a)
What is the mean of $ \mathbf{Y\left(t\right)} $ ?
(b)
What is the autocorrelation function of $ \mathbf{Y}\left(t\right) $ ?
(c)
Write an expression for the $ n $ -th order characteristic function of $ \mathbf{Y}\left(t\right) $ sampled at time $ t_{1},t_{2},\cdots,t_{n} $ . Simplify as much as possible.
(d)
Write an expression for the second-order pdf $ f_{\mathbf{Y}\left(t_{1}\right)\mathbf{Y}\left(t_{2}\right)}\left(y_{1},y_{2}\right) $ of $ \mathbf{Y}\left(t\right) $ . simplify as much as possible.
(e)
Find the minium mean-square estimate of $ \mathbf{Y}\left(t_{2}\right) $ given that $ \mathbf{Y}\left(t_{1}\right)=y_{1} $ . Simplify your answer as much as possible.
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