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 $b$ .

$Var\left[\mathbf{Z}\right]=a^{2}\sigma_{\mathbf{X}}^{2}+2rab\sigma_{\mathbf{X}}\sigma_{\mathbf{Y}}+b^{2}\sigma_{\mathbf{Y}}^{2}.$