(New page: ==7.6 QE 2003 January== '''Problem 1 (30 points)''' '''i)''' Let <math>\mathbf{X}</math> and <math>\mathbf{Y}</math> be jointly Gaussian (normal) distributed random variables with mea...) |
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'''i)''' | '''i)''' | ||
− | Let <math>\mathbf{X}</math> and <math>\mathbf{Y}</math> be jointly Gaussian (normal) distributed random variables with mean <math>0</math> , <math>E\left[\mathbf{X}^{2}\right]=E\left[\mathbf{Y}^{2}\right]=\sigma^{2}</math> and <math>E\left[\mathbf{XY}\right]=\rho\sigma^{2}</math> with <math>\left|\rho\right|<1</math> . Find the joint characteristic function <math>E\left[e^{i\left(h_{1}\mathbf{X}+h_{2}\mathbf{Y}\right)}\right]</math> . | + | Let <math class="inline">\mathbf{X}</math> and <math class="inline">\mathbf{Y}</math> be jointly Gaussian (normal) distributed random variables with mean <math class="inline">0</math> , <math class="inline">E\left[\mathbf{X}^{2}\right]=E\left[\mathbf{Y}^{2}\right]=\sigma^{2}</math> and <math class="inline">E\left[\mathbf{XY}\right]=\rho\sigma^{2}</math> with <math class="inline">\left|\rho\right|<1</math> . Find the joint characteristic function <math class="inline">E\left[e^{i\left(h_{1}\mathbf{X}+h_{2}\mathbf{Y}\right)}\right]</math> . |
− | • We can find the correlation coefficient using the covariance and variances <math>r=\frac{Cov\left(\mathbf{X},\mathbf{Y}\right)}{\sigma^{2}}=\frac{E\left[\mathbf{XY}\right]-E\left[\mathbf{X}\right]E\left[\mathbf{Y}\right]}{\sigma^{2}}=\frac{\rho\sigma^{2}-0\cdot0}{\sigma^{2}}=\rho.</math> | + | • We can find the correlation coefficient using the covariance and variances <math class="inline">r=\frac{Cov\left(\mathbf{X},\mathbf{Y}\right)}{\sigma^{2}}=\frac{E\left[\mathbf{XY}\right]-E\left[\mathbf{X}\right]E\left[\mathbf{Y}\right]}{\sigma^{2}}=\frac{\rho\sigma^{2}-0\cdot0}{\sigma^{2}}=\rho.</math> |
− | • Now, we can get the joint characteristic function <math>\Phi_{\mathbf{X}\mathbf{Y}}\left(\omega_{1},\omega_{2}\right)=e^{i\left(0\cdot\omega_{1}+0\cdot\omega_{2}\right)}e^{-\frac{1}{2}\left(\sigma^{2}\omega_{1}^{2}+2r\sigma^{2}\omega_{1}\omega_{2}+\sigma^{2}\omega_{2}\right)}=e^{-\frac{1}{2}\sigma^{2}\omega^{2}\left(2+2\rho\right)}=e^{-\sigma^{2}\omega^{2}\left(1+\rho\right)}.</math> | + | • Now, we can get the joint characteristic function <math class="inline">\Phi_{\mathbf{X}\mathbf{Y}}\left(\omega_{1},\omega_{2}\right)=e^{i\left(0\cdot\omega_{1}+0\cdot\omega_{2}\right)}e^{-\frac{1}{2}\left(\sigma^{2}\omega_{1}^{2}+2r\sigma^{2}\omega_{1}\omega_{2}+\sigma^{2}\omega_{2}\right)}=e^{-\frac{1}{2}\sigma^{2}\omega^{2}\left(2+2\rho\right)}=e^{-\sigma^{2}\omega^{2}\left(1+\rho\right)}.</math> |
'''ii)''' | '''ii)''' | ||
− | Let <math>\mathbf{X}</math> and <math>\mathbf{Y}</math> be two jointly Gaussian distributed r.v's with identical means and variances but are not necessarily independent. Show that the r.v. <math>\mathbf{V}=\mathbf{X}+\mathbf{Y}</math> is independeent of the r.v. <math>\mathbf{W}=\mathbf{X}-\mathbf{Y}</math> . Is the same answer true for <math>\mathbf{A}=f\left(\mathbf{V}\right)</math> and <math>\mathbf{B}=g\left(\mathbf{W}\right)</math> where <math>f\left(\cdot\right)</math> and <math>g\left(\cdot\right)</math> are suitable functions such that <math>E\left[f\left(\mathbf{V}\right)\right]<\infty</math> and <math>E\left[g\left(\mathbf{W}\right)\right]<\infty</math> . Given reasons. | + | Let <math class="inline">\mathbf{X}</math> and <math class="inline">\mathbf{Y}</math> be two jointly Gaussian distributed r.v's with identical means and variances but are not necessarily independent. Show that the r.v. <math class="inline">\mathbf{V}=\mathbf{X}+\mathbf{Y}</math> is independeent of the r.v. <math class="inline">\mathbf{W}=\mathbf{X}-\mathbf{Y}</math> . Is the same answer true for <math class="inline">\mathbf{A}=f\left(\mathbf{V}\right)</math> and <math class="inline">\mathbf{B}=g\left(\mathbf{W}\right)</math> where <math class="inline">f\left(\cdot\right)</math> and <math class="inline">g\left(\cdot\right)</math> are suitable functions such that <math class="inline">E\left[f\left(\mathbf{V}\right)\right]<\infty</math> and <math class="inline">E\left[g\left(\mathbf{W}\right)\right]<\infty</math> . Given reasons. |
'''iii)''' | '''iii)''' | ||
− | Let <math>\mathbf{X}</math> and <math>\mathbf{Y}</math> be independent <math>N\left(m,1\right)</math> random variables. Show that the sample mean <math>\mathbf{M}=\frac{\mathbf{X}+\mathbf{Y}}{2}</math> is independent of the sample variance <math>\mathbf{V}=\left(\mathbf{X}-\mathbf{M}\right)^{2}+\left(\mathbf{Y}-\mathbf{M}\right)^{2}</math> . Note: <math>\mathbf{V}</math> is not a Gaussian random variable. | + | Let <math class="inline">\mathbf{X}</math> and <math class="inline">\mathbf{Y}</math> be independent <math class="inline">N\left(m,1\right)</math> random variables. Show that the sample mean <math class="inline">\mathbf{M}=\frac{\mathbf{X}+\mathbf{Y}}{2}</math> is independent of the sample variance <math class="inline">\mathbf{V}=\left(\mathbf{X}-\mathbf{M}\right)^{2}+\left(\mathbf{Y}-\mathbf{M}\right)^{2}</math> . Note: <math class="inline">\mathbf{V}</math> is not a Gaussian random variable. |
'''Problem 2 (35 points)''' | '''Problem 2 (35 points)''' | ||
− | Consider the stochastic process <math>\left\{ \mathbf{X}_{n}\right\}</math> defined by: <math>\mathbf{X}_{n+1}=a\mathbf{X}_{n}+b\mathbf{W}_{n} where \mathbf{X}_{0}\sim N\left(0,\sigma^{2}\right)</math> and <math>\left\{ \mathbf{W}_{n}\right\}</math> is an i.i.d. <math>N\left(0,1\right)</math> sequence of r.v's independent of <math>\mathbf{X}_{0}</math> . | + | Consider the stochastic process <math class="inline">\left\{ \mathbf{X}_{n}\right\}</math> defined by: <math class="inline">\mathbf{X}_{n+1}=a\mathbf{X}_{n}+b\mathbf{W}_{n} where \mathbf{X}_{0}\sim N\left(0,\sigma^{2}\right)</math> and <math class="inline">\left\{ \mathbf{W}_{n}\right\}</math> is an i.i.d. <math class="inline">N\left(0,1\right)</math> sequence of r.v's independent of <math class="inline">\mathbf{X}_{0}</math> . |
'''i)''' | '''i)''' | ||
− | Show that if <math>R_{k}=cov\left(\mathbf{X}_{k},\mathbf{X}_{k}\right)</math> converges as <math>k\rightarrow\infty</math> , then <math>\left\{ \mathbf{X}_{k}\right\}</math> converges to a w.s.s. process. | + | Show that if <math class="inline">R_{k}=cov\left(\mathbf{X}_{k},\mathbf{X}_{k}\right)</math> converges as <math class="inline">k\rightarrow\infty</math> , then <math class="inline">\left\{ \mathbf{X}_{k}\right\}</math> converges to a w.s.s. process. |
'''ii)''' | '''ii)''' | ||
− | Show that if <math>\sigma^{2}</math> is chosen appropriately and <math>\left|a\right|<1</math> , then <math>\left\{ \mathbf{X}_{k}\right\}</math> will be a stationary process for all <math>k</math> . | + | Show that if <math class="inline">\sigma^{2}</math> is chosen appropriately and <math class="inline">\left|a\right|<1</math> , then <math class="inline">\left\{ \mathbf{X}_{k}\right\}</math> will be a stationary process for all <math class="inline">k</math> . |
'''iii)''' | '''iii)''' | ||
− | If <math>\left|a\right|>1</math> , show that the variance of the process <math>\left\{ \mathbf{X}_{k}\right\}</math> diverges but <math>\frac{\mathbf{X}_{k}}{\left|a\right|^{k}}</math> converges in the mean square. | + | If <math class="inline">\left|a\right|>1</math> , show that the variance of the process <math class="inline">\left\{ \mathbf{X}_{k}\right\}</math> diverges but <math class="inline">\frac{\mathbf{X}_{k}}{\left|a\right|^{k}}</math> converges in the mean square. |
Problem 3 (35 points) | Problem 3 (35 points) | ||
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'''i)''' | '''i)''' | ||
− | Catastrophes occur at times <math>\mathbf{T}_{1},\mathbf{T}_{2},\cdots</math>, where <math>\mathbf{T}_{i}=\sum_{k=1}^{i}\mathbf{X}_{k}</math> where the <math>\mathbf{X}_{k}</math> 's are independent, identically distributed positive random variables. Let <math>\mathbf{N}_{t}=\max\left\{ n:\mathbf{T}_{n}\leq t\right\}</math> be the number of catastrophes which have occurred by time <math>t</math> . Show that if <math>E\left[\mathbf{X}_{1}\right]<\infty</math> then <math>\mathbf{N}_{t}\rightarrow\infty</math> almost surely (a.s.) and <math>\frac{\mathbf{N}_{t}}{t}\rightarrow\frac{1}{E\left[\mathbf{X}_{1}\right]}</math> as <math>t\rightarrow\infty</math> a.s. | + | Catastrophes occur at times <math class="inline">\mathbf{T}_{1},\mathbf{T}_{2},\cdots</math>, where <math class="inline">\mathbf{T}_{i}=\sum_{k=1}^{i}\mathbf{X}_{k}</math> where the <math class="inline">\mathbf{X}_{k}</math> 's are independent, identically distributed positive random variables. Let <math class="inline">\mathbf{N}_{t}=\max\left\{ n:\mathbf{T}_{n}\leq t\right\}</math> be the number of catastrophes which have occurred by time <math class="inline">t</math> . Show that if <math class="inline">E\left[\mathbf{X}_{1}\right]<\infty</math> then <math class="inline">\mathbf{N}_{t}\rightarrow\infty</math> almost surely (a.s.) and <math class="inline">\frac{\mathbf{N}_{t}}{t}\rightarrow\frac{1}{E\left[\mathbf{X}_{1}\right]}</math> as <math class="inline">t\rightarrow\infty</math> a.s. |
'''ii)''' | '''ii)''' | ||
− | Let <math>\left\{ \mathbf{X}_{t},t\geq0\right\}</math> be a stochastic process defined by: <math>\mathbf{X}_{t}=\sqrt{2}\cos\left(2\pi\xi t\right)</math> where <math>\xi</math> is a <math>N\left(0,1\right)</math> random variable. Show that as <math>t\rightarrow\infty,\;\left\{ \mathbf{X}_{t}\right\}</math> converges to a wide sense stationary process. Find the spectral density of the limit process. | + | Let <math class="inline">\left\{ \mathbf{X}_{t},t\geq0\right\}</math> be a stochastic process defined by: <math class="inline">\mathbf{X}_{t}=\sqrt{2}\cos\left(2\pi\xi t\right)</math> where <math class="inline">\xi</math> is a <math class="inline">N\left(0,1\right)</math> random variable. Show that as <math class="inline">t\rightarrow\infty,\;\left\{ \mathbf{X}_{t}\right\}</math> converges to a wide sense stationary process. Find the spectral density of the limit process. |
'''Hint:''' | '''Hint:''' | ||
− | Use the fact that the characteristic function of a <math>N\left(0,1\right)</math> r.v. is given by <math>E\left[e^{ih\mathbf{X}}\right]=e^{-\frac{h^{2}}{2}}</math> . | + | Use the fact that the characteristic function of a <math class="inline">N\left(0,1\right)</math> r.v. is given by <math class="inline">E\left[e^{ih\mathbf{X}}\right]=e^{-\frac{h^{2}}{2}}</math> . |
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Revision as of 07:27, 1 December 2010
7.6 QE 2003 January
Problem 1 (30 points)
i)
Let $ \mathbf{X} $ and $ \mathbf{Y} $ be jointly Gaussian (normal) distributed random variables with mean $ 0 $ , $ E\left[\mathbf{X}^{2}\right]=E\left[\mathbf{Y}^{2}\right]=\sigma^{2} $ and $ E\left[\mathbf{XY}\right]=\rho\sigma^{2} $ with $ \left|\rho\right|<1 $ . Find the joint characteristic function $ E\left[e^{i\left(h_{1}\mathbf{X}+h_{2}\mathbf{Y}\right)}\right] $ .
• We can find the correlation coefficient using the covariance and variances $ r=\frac{Cov\left(\mathbf{X},\mathbf{Y}\right)}{\sigma^{2}}=\frac{E\left[\mathbf{XY}\right]-E\left[\mathbf{X}\right]E\left[\mathbf{Y}\right]}{\sigma^{2}}=\frac{\rho\sigma^{2}-0\cdot0}{\sigma^{2}}=\rho. $
• Now, we can get the joint characteristic function $ \Phi_{\mathbf{X}\mathbf{Y}}\left(\omega_{1},\omega_{2}\right)=e^{i\left(0\cdot\omega_{1}+0\cdot\omega_{2}\right)}e^{-\frac{1}{2}\left(\sigma^{2}\omega_{1}^{2}+2r\sigma^{2}\omega_{1}\omega_{2}+\sigma^{2}\omega_{2}\right)}=e^{-\frac{1}{2}\sigma^{2}\omega^{2}\left(2+2\rho\right)}=e^{-\sigma^{2}\omega^{2}\left(1+\rho\right)}. $
ii)
Let $ \mathbf{X} $ and $ \mathbf{Y} $ be two jointly Gaussian distributed r.v's with identical means and variances but are not necessarily independent. Show that the r.v. $ \mathbf{V}=\mathbf{X}+\mathbf{Y} $ is independeent of the r.v. $ \mathbf{W}=\mathbf{X}-\mathbf{Y} $ . Is the same answer true for $ \mathbf{A}=f\left(\mathbf{V}\right) $ and $ \mathbf{B}=g\left(\mathbf{W}\right) $ where $ f\left(\cdot\right) $ and $ g\left(\cdot\right) $ are suitable functions such that $ E\left[f\left(\mathbf{V}\right)\right]<\infty $ and $ E\left[g\left(\mathbf{W}\right)\right]<\infty $ . Given reasons.
iii)
Let $ \mathbf{X} $ and $ \mathbf{Y} $ be independent $ N\left(m,1\right) $ random variables. Show that the sample mean $ \mathbf{M}=\frac{\mathbf{X}+\mathbf{Y}}{2} $ is independent of the sample variance $ \mathbf{V}=\left(\mathbf{X}-\mathbf{M}\right)^{2}+\left(\mathbf{Y}-\mathbf{M}\right)^{2} $ . Note: $ \mathbf{V} $ is not a Gaussian random variable.
Problem 2 (35 points)
Consider the stochastic process $ \left\{ \mathbf{X}_{n}\right\} $ defined by: $ \mathbf{X}_{n+1}=a\mathbf{X}_{n}+b\mathbf{W}_{n} where \mathbf{X}_{0}\sim N\left(0,\sigma^{2}\right) $ and $ \left\{ \mathbf{W}_{n}\right\} $ is an i.i.d. $ N\left(0,1\right) $ sequence of r.v's independent of $ \mathbf{X}_{0} $ .
i)
Show that if $ R_{k}=cov\left(\mathbf{X}_{k},\mathbf{X}_{k}\right) $ converges as $ k\rightarrow\infty $ , then $ \left\{ \mathbf{X}_{k}\right\} $ converges to a w.s.s. process.
ii)
Show that if $ \sigma^{2} $ is chosen appropriately and $ \left|a\right|<1 $ , then $ \left\{ \mathbf{X}_{k}\right\} $ will be a stationary process for all $ k $ .
iii)
If $ \left|a\right|>1 $ , show that the variance of the process $ \left\{ \mathbf{X}_{k}\right\} $ diverges but $ \frac{\mathbf{X}_{k}}{\left|a\right|^{k}} $ converges in the mean square.
Problem 3 (35 points)
i)
Catastrophes occur at times $ \mathbf{T}_{1},\mathbf{T}_{2},\cdots $, where $ \mathbf{T}_{i}=\sum_{k=1}^{i}\mathbf{X}_{k} $ where the $ \mathbf{X}_{k} $ 's are independent, identically distributed positive random variables. Let $ \mathbf{N}_{t}=\max\left\{ n:\mathbf{T}_{n}\leq t\right\} $ be the number of catastrophes which have occurred by time $ t $ . Show that if $ E\left[\mathbf{X}_{1}\right]<\infty $ then $ \mathbf{N}_{t}\rightarrow\infty $ almost surely (a.s.) and $ \frac{\mathbf{N}_{t}}{t}\rightarrow\frac{1}{E\left[\mathbf{X}_{1}\right]} $ as $ t\rightarrow\infty $ a.s.
ii)
Let $ \left\{ \mathbf{X}_{t},t\geq0\right\} $ be a stochastic process defined by: $ \mathbf{X}_{t}=\sqrt{2}\cos\left(2\pi\xi t\right) $ where $ \xi $ is a $ N\left(0,1\right) $ random variable. Show that as $ t\rightarrow\infty,\;\left\{ \mathbf{X}_{t}\right\} $ converges to a wide sense stationary process. Find the spectral density of the limit process.
Hint:
Use the fact that the characteristic function of a $ N\left(0,1\right) $ r.v. is given by $ E\left[e^{ih\mathbf{X}}\right]=e^{-\frac{h^{2}}{2}} $ .