Example. Addition of two jointly distributed Gaussian random variables
Let $ \mathbf{X} $ and $ \mathbf{Y} $ be two jointly distributed Gaussian random variables. Assume $ \mathbf{X} $ has mean $ \mu_{\mathbf{X}} $ and variance $ \sigma_{\mathbf{X}}^{2} , \mathbf{Y} $ has mean $ \mu_{\mathbf{Y}} $ and variance $ \sigma_{\mathbf{Y}}^{2} $ , and that the correlation coefficient between $ \mathbf{X} $ and $ \mathbf{Y} $ is $ r $ . Define a new random variable $ \mathbf{Z}=\mathbf{X}+\mathbf{Y} $ .
(a)
Show that $ \mathbf{Z} $ is a Gaussian random variable.
If $ \mathbf{Z} $ is a Guassian random variable, then it has a characteristic function of the form
$ \Phi_{\mathbf{Z}}\left(\omega\right)=e^{i\mu_{\mathbf{Z}}\omega}e^{-\frac{1}{2}\sigma_{\mathbf{Z}}^{2}\omega^{2}}. $
$ \Phi_{\mathbf{Z}}\left(\omega\right) $
where $ \Phi_{\mathbf{XY}}\left(\omega_{1},\omega_{2}\right) $ is the joint characteristic function of $ \mathbf{X} $ and $ \mathbf{Y} $ , defined as
$ \Phi_{\mathbf{XY}}\left(\omega_{1},\omega_{2}\right)=E\left[e^{i\left(\mathbf{\omega_{1}X}+\omega_{2}\mathbf{Y}\right)}\right]. $
Now because $ \mathbf{X} $ and $ \mathbf{Y} $ are jointly Gaussian with the given parameters, we know that
$ \Phi_{\mathbf{XY}}\left(\omega_{1},\omega_{2}\right)=e^{i\left(\mu_{X}\omega_{1}+\mu_{Y}\omega_{2}\right)}e^{-\frac{1}{2}\left(\sigma_{X}^{2}\omega_{1}^{2}+2r\sigma_{X}\sigma_{Y}\omega_{1}\omega_{2}+\sigma_{Y}^{2}\omega_{2}^{2}\right)}. $
Thus,
$ \Phi_{\mathbf{Z}}\left(\omega\right)=\Phi_{\mathbf{XY}}\left(\omega,\omega\right)=e^{i\left(\mu_{X}\omega+\mu_{Y}\omega\right)}e^{-\frac{1}{2}\left(\sigma_{X}^{2}\omega^{2}+2r\sigma_{X}\sigma_{Y}\omega^{2}+\sigma_{Y}^{2}\omega^{2}\right)} $$ =e^{i\left(\mu_{X}+\mu_{Y}\right)\omega}e^{-\frac{1}{2}\left(\sigma_{X}^{2}+2r\sigma_{X}\sigma_{Y}+\sigma_{Y}^{2}\right)\omega^{2}}=e^{i\mu_{Z}\omega}e^{-\frac{1}{2}\sigma_{Z}^{2}\omega^{2}} $
where $ \mu_{Z}=\mu_{X}+\mu_{Y} $ and $ \sigma_{Z}^{2}=\sigma_{X}^{2}+2r\sigma_{X}\sigma_{Y}+\sigma_{Y}^{2} $ .
$ \mathbf{Z} $ is a Gaussian random variable with $ E\left[\mathbf{Z}\right]=\mu_{X}+\mu_{Y} and Var\left[\mathbf{Z}\right]=\sigma_{X}^{2}+2r\sigma_{X}\sigma_{Y}+\sigma_{Y}^{2} $ .
(b)
Find the variance of $ \mathbf{Z} $ .
As show in part (a) $ Var\left[\mathbf{Z}\right]=\sigma_{\mathbf{X}}^{2}+2r\sigma_{\mathbf{X}}\sigma_{\mathbf{Y}}+\sigma_{\mathbf{Y}}^{2} $ .
