Communication, Networking, Signal and Image Processing (CS)
Question 1: Probability and Random Processes
August 2008
3
Consider a random process $ \mathbf{X}\left(t\right) $ that assumes values $ \pm1 $ . Suppose that $ \mathbf{X}\left(0\right)=\pm1 $ with probability $ 1/2 $ , and suppose that $ \mathbf{X}\left(t\right) $ then changes polarity with each occurrence of an event in a Poisson process of rate $ \lambda $ .
Note:
You might find the equations $ \frac{1}{2}\left(e^{x}+e^{-x}\right)=\sum_{j=0}^{\infty}\frac{x^{2j}}{\left(2j\right)!} $ and $ \frac{1}{2}\left(e^{x}-e^{-x}\right)=\sum_{j=0}^{\infty}\frac{x^{2j+1}}{\left(2j+1\right)!} $ helpful.
a. (15 points)
Find the probability mass function of $ \mathbf{X}\left(t\right) $ .
$ P\left(\left\{ \mathbf{X}\left(t\right)=1\right\} |\left\{ \mathbf{X}\left(0\right)=1\right\} \right)=P\left(\left\{ \mathbf{N}\left(t\right)=\text{even}\right\} \right)=\sum_{n=0}^{\infty}e^{-\lambda t}\frac{\left(\lambda t\right)^{2n}}{\left(2n\right)!}=e^{-\lambda t}\sum_{n=0}^{\infty}\frac{\left(\lambda t\right)^{2n}}{\left(2n\right)!} $$ =e^{-\lambda t}\cdot\frac{1}{2}\left(e^{\lambda t}+e^{-\lambda t}\right)=\frac{1}{2}\left(1+e^{-2\lambda t}\right). $
$ P\left(\left\{ \mathbf{X}\left(t\right)=1\right\} |\left\{ \mathbf{X}\left(0\right)=-1\right\} \right)=P\left(\left\{ \mathbf{N}\left(t\right)=\text{odd}\right\} \right)=\sum_{n=0}^{\infty}e^{-\lambda t}\frac{\left(\lambda t\right)^{2n+1}}{\left(2n+1\right)!}=e^{-\lambda t}\sum_{n=0}^{\infty}\frac{\left(\lambda t\right)^{2n+1}}{\left(2n+1\right)!} $$ =e^{-\lambda t}\cdot\frac{1}{2}\left(e^{\lambda t}-e^{-\lambda t}\right)=\frac{1}{2}\left(1-e^{-2\lambda t}\right). $
$ P\left(\left\{ \mathbf{X}\left(t\right)=1\right\} \right)=P\left(\left\{ \mathbf{X}\left(t\right)=1\right\} |\left\{ \mathbf{X}\left(0\right)=1\right\} \right)P\left(\left\{ \mathbf{X}\left(0\right)=1\right\} \right)+P\left(\left\{ \mathbf{X}\left(t\right)=1\right\} |\left\{ \mathbf{X}\left(0\right)=-1\right\} \right)P\left(\left\{ \mathbf{X}\left(0\right)=-1\right\} \right) $$ =\frac{1}{2}\left(1+e^{-2\lambda t}\right)\cdot\frac{1}{2}+\frac{1}{2}\left(1-e^{-2\lambda t}\right)\cdot\frac{1}{2}=\frac{1}{2}. $
Thus, $ P\left(\left\{ \mathbf{X}\left(t\right)=1\right\} \right)=P\left(\left\{ \mathbf{X}\left(t\right)=-1\right\} \right)=\frac{1}{2} $ .
b. (15 points)
Find the autocovariance function of the random process $ \mathbf{X}\left(t\right) $ .
Note
The explanation about the autocovariance function is here.
$ E\left[\mathbf{X}\left(t\right)\right]=1\cdot\frac{1}{2}+\left(-1\right)\cdot\frac{1}{2}=-\frac{1}{2}+\frac{1}{2}=0. $
$ C_{\mathbf{XX}}\left(t_{1},t_{2}\right)=E\left[\left(\mathbf{X}\left(t_{1}\right)-\mu_{\mathbf{X}}\left(t_{1}\right)\right)\left(\mathbf{X}\left(t_{2}\right)-\mu_{\mathbf{X}}\left(t_{2}\right)\right)^{*}\right]=R_{\mathbf{XX}}\left(t_{1},t_{2}\right)-\mu_{\mathbf{X}}\left(t_{1}\right)\mu_{\mathbf{X}}\left(t_{2}\right)^{*} $$ =R_{\mathbf{XX}}\left(t_{1},t_{2}\right)=1\cdot P\left(\left\{ \mathbf{X}\left(t_{1}=1\right)\cap\mathbf{X}\left(t_{2}=1\right)\right\} \right)-1\cdot P\left(\left\{ \mathbf{X}\left(t_{1}=-1\right)\cap\mathbf{X}\left(t_{2}=1\right)\right\} \right) $$ -1\cdot P\left(\left\{ \mathbf{X}\left(t_{1}=1\right)\cap\mathbf{X}\left(t_{2}=-1\right)\right\} \right)+1\cdot P\left(\left\{ \mathbf{X}\left(t_{1}=-1\right)\cap\mathbf{X}\left(t_{2}=-1\right)\right\} \right) $$ =1\cdot P\left(\left\{ \mathbf{X}\left(t_{1}\right)=\mathbf{X}\left(t_{2}\right)\right\} \right)-1\cdot P\left(\left\{ \mathbf{X}\left(t_{1}\right)\neq\mathbf{X}\left(t_{2}\right)\right\} \right). $
$ P\left(\left\{ \mathbf{X}\left(t_{1}\right)=\mathbf{X}\left(t_{2}\right)\right\} \right)=P\left(\left\{ \text{number of events occuring within }\left(t_{1},t_{2}\right)\text{is even}\right\} \right) $
$ =\sum_{n=0}^{\infty}e^{-\lambda\left(t_{2}-t_{1}\right)}\frac{\left[\lambda\left(t_{2}-t_{1}\right)\right]^{2n}}{\left(2n\right)!}=\frac{1}{2}e^{-\lambda\left(t_{2}-t_{1}\right)}\left(e^{\lambda\left(t_{2}-t_{1}\right)}+e^{-\lambda\left(t_{2}-t_{1}\right)}\right) $$ =\frac{1}{2}\left(1+e^{-2\lambda\left(t_{2}-t_{1}\right)}\right). $
By using the same method above, we can get $ P\left(\left\{ \mathbf{X}\left(t_{1}\right)\neq\mathbf{X}\left(t_{2}\right)\right\} \right)=\frac{1}{2}\left(1-e^{-2\lambda\left(t_{2}-t_{1}\right)}\right) $ . Thus,
$ C_{\mathbf{XX}}\left(t_{1},t_{2}\right)=\frac{1}{2}\left(1+e^{-2\lambda\left(t_{2}-t_{1}\right)}\right)-\frac{1}{2}\left(1-e^{-2\lambda\left(t_{2}-t_{1}\right)}\right)=e^{-2\lambda\left(t_{2}-t_{1}\right)}. $
