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Let <math class="inline">\mathbf{X}</math> be a random variable with absolutely continuous probability distribution function. Show that for any <math class="inline">\alpha>0</math> and any real number <math class="inline">s</math> :<math class="inline">P\left(e^{s\mathbf{X}}\geq\alpha\right)\leq\frac{\phi\left(s\right)}{\alpha}</math> where <math class="inline">\phi\left(s\right)</math> is the moment generating function, <math class="inline">\phi\left(s\right)=E\left[e^{s\mathbf{X}}\right]</math> . Note: <math class="inline">\phi\left(s\right)</math> can be related to the Laplace Transform of <math class="inline">f_{\mathbf{X}}\left(x\right)</math> . | Let <math class="inline">\mathbf{X}</math> be a random variable with absolutely continuous probability distribution function. Show that for any <math class="inline">\alpha>0</math> and any real number <math class="inline">s</math> :<math class="inline">P\left(e^{s\mathbf{X}}\geq\alpha\right)\leq\frac{\phi\left(s\right)}{\alpha}</math> where <math class="inline">\phi\left(s\right)</math> is the moment generating function, <math class="inline">\phi\left(s\right)=E\left[e^{s\mathbf{X}}\right]</math> . Note: <math class="inline">\phi\left(s\right)</math> can be related to the Laplace Transform of <math class="inline">f_{\mathbf{X}}\left(x\right)</math> . | ||
Latest revision as of 10:16, 10 March 2015
Communication, Networking, Signal and Image Processing (CS)
Question 1: Probability and Random Processes
January 2002
Question
1. (20 pts)
Given two coins; the first coin is fair and the second coin has two heads. One coin is picked at random and tossed two times. It shows heads both times. What is the probability that the coin picked is fair?
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2. (20 pts)
Let $ \mathbf{X}_{t} $ and $ \mathbf{Y}_{t} $ by jointly wide sense stationary continous parameter random processes with $ E\left[\left|\mathbf{X}\left(0\right)-\mathbf{Y}\left(0\right)\right|^{2}\right]=0 $ . Show that $ R_{\mathbf{X}}\left(\tau\right)=R_{\mathbf{Y}}\left(\tau\right)=R_{\mathbf{XY}}\left(\tau\right) $ .
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3. (20 pts)
Let $ \mathbf{X}_{t} $ be a zero mean continuous parameter random process. Let $ g(t) $ and $ w\left(t\right) $ be measurable functions defined on the real numbers. Further, let $ w\left(t\right) $ be even. Let the autocorrelation function of $ \mathbf{X}_{t} $ be $ \frac{g\left(t_{1}\right)g\left(t_{2}\right)}{w\left(t_{1}-t_{2}\right)} $ . From the new random process $ \mathbf{Y}_{i}=\frac{\mathbf{X}\left(t\right)}{g\left(t\right)} $ . Is $ \mathbf{Y}_{t} $ w.s.s. ?
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4. (20 pts)
Let $ \mathbf{X}_{1},\mathbf{X}_{2},\cdots,\mathbf{X}_{n} $ be i.i.d. random variables with absolutely continuous probability distribution function $ F\left(x\right) $ . Let the random variable $ \mathbf{Y}_{j} $ be the $ j $ -th order statistic of the $ \mathbf{X}_{i} $ 's. that is: $ \mathbf{Y}_{j}=j\text{-th smallest of }\left\{ \mathbf{X}_{1},\mathbf{X}_{2},\cdots,\mathbf{X}_{n}\right\} $ .
(a)
What is another name for the first order statistic?
(b)
What is another name for the n/2 order statistic?
(c)
Find the probability density function of the first order statistic. (You may assume n is odd.)
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5. (20 pts)
Let $ \mathbf{X} $ be a random variable with absolutely continuous probability distribution function. Show that for any $ \alpha>0 $ and any real number $ s $ :$ P\left(e^{s\mathbf{X}}\geq\alpha\right)\leq\frac{\phi\left(s\right)}{\alpha} $ where $ \phi\left(s\right) $ is the moment generating function, $ \phi\left(s\right)=E\left[e^{s\mathbf{X}}\right] $ . Note: $ \phi\left(s\right) $ can be related to the Laplace Transform of $ f_{\mathbf{X}}\left(x\right) $ .
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