Communication, Networking, Signal and Image Processing (CS)
Question 1: Probability and Random Processes
August 2002
2. (25 Points)
Let $ \mathbf{X} $ and $ \mathbf{Y} $ be two jointly distributed Gaussian random variables. The random variable $ \mathbf{X} $ has mean $ \mu_{\mathbf{X}} $ and variance $ \sigma_{\mathbf{X}}^{2} $ . The correlation coefficient between $ \mathbf{X} $ and $ \mathbf{Y} $ is $ r $ . Define a new random variable $ \mathbf{Z} $ by $ \mathbf{Z}=a\mathbf{X}+b\mathbf{Y} $, where $ a $ and $ b $ are real numbers.
(a)
Prove that $ \mathbf{Z} $ is a Gaussian random variable.
$ \Phi_{\mathbf{Z}}\left(\omega\right)=E\left[e^{i\omega\mathbf{Z}}\right]=E\left[e^{i\omega\left(a\mathbf{X}+b\mathbf{Y}\right)}\right]=\Phi_{\mathbf{XY}}\left(a\omega,b\omega\right). $
$ \Phi_{\mathbf{XY}}\left(\omega_{1},\omega_{2}\right)=\exp\left[i\left(\mu_{\mathbf{X}}\omega_{1}+\mu_{\mathbf{Y}}\omega_{2}\right)-\frac{1}{2}\left(\sigma_{\mathbf{X}}^{2}\omega_{1}^{2}+2r\sigma_{\mathbf{X}}\sigma_{\mathbf{Y}}\omega_{1}\omega_{2}+\sigma_{\mathbf{Y}}^{2}\omega_{2}^{2}\right)\right]. $
$ \Phi_{\mathbf{Z}}\left(\omega\right)=\Phi_{\mathbf{XY}}\left(a\omega,b\omega\right)=\exp\left[i\left(a\mu_{\mathbf{X}}+b\mu_{\mathbf{Y}}\right)\omega-\frac{1}{2}\left(a^{2}\sigma_{\mathbf{X}}^{2}+2rab\sigma_{\mathbf{X}}\sigma_{\mathbf{Y}}+b^{2}\sigma_{\mathbf{Y}}^{2}\right)\omega^{2}\right], $
which is the characteristic function of a Gaussian random variable with mean $ a\mu_{\mathbf{X}}+b\mu_{\mathbf{Y}} $ and variance $ a^{2}\sigma_{\mathbf{X}}^{2}+2rab\sigma_{\mathbf{X}}\sigma_{\mathbf{Y}}+b^{2}\sigma_{\mathbf{Y}}^{2} $ .
(b)
Find the mean of $ \mathbf{Z} $ . Express your answer in terms of the parameters $ \mu_{\mathbf{X}} $ , $ \sigma_{\mathbf{X}}^{2} $ , $ \mu_{\mathbf{Y}} $ , $ \sigma_{\mathbf{Y}}^{2} $ , $ r $ , $ a $ , and $ b $ .
$ E\left[\mathbf{Z}\right]=a\mu_{\mathbf{X}}+b\mu_{\mathbf{Y}}. $
(c)
Find the variance of $ \mathbf{Z} $ . Express your answer in terms of the parameters $ \mu_{\mathbf{X}} $ , $ \sigma_{\mathbf{X}}^{2} $ , $ \mu_{\mathbf{Y}} $ , $ \sigma_{\mathbf{Y}}^{2} $ , $ r $ , $ a $ , and $ b $ .
$ Var\left[\mathbf{Z}\right]=a^{2}\sigma_{\mathbf{X}}^{2}+2rab\sigma_{\mathbf{X}}\sigma_{\mathbf{Y}}+b^{2}\sigma_{\mathbf{Y}}^{2}. $
