Example. Two jointly distributed random variables (Joint characteristic function)

Let $ \mathbf{X} $ and $ \mathbf{Y} $ be tweo jointly distributed random variables having joint characteristic function

$ \Phi_{\mathbf{XY}}\left(\omega_{1},\omega_{2}\right)=\frac{1}{\left(1-i\omega_{1}\right)\left(1-i\omega_{2}\right)}. $

(a) Calculate $ E\left[\mathbf{X}\right] $ .

$ \Phi_{\mathbf{X}}\left(\omega\right)=\Phi_{\mathbf{XY}}\left(\omega,0\right)=\frac{1}{1-i\omega}=\left(1-i\omega\right)^{-1} $

$ E\left[\mathbf{X}\right]=\frac{d}{d\left(i\omega\right)}\Phi_{\mathbf{X}}\left(\omega\right)|_{i\omega=0}=(-1)(1-i\omega)^{-2}(-1)|_{i\omega=0}=1 $

(b) Calculate $ E\left[\mathbf{Y}\right] $

$ E\left[\mathbf{Y}\right]=1 $

(c) Calculate $ E\left[\mathbf{XY}\right] $ .

$ E\left[\mathbf{XY}\right]=\frac{\partial^{2}}{\partial\left(i\omega_{1}\right)\partial\left(i\omega_{2}\right)}\Phi_{\mathbf{XY}}\left(\omega_{1},\omega_{2}\right)|_{i\omega_{1}=i\omega_{2}=0}=\left(1-i\omega_{1}\right)^{-2}\left(1-i\omega_{2}\right)^{-2}|_{i\omega_{1}=i\omega_{2}=0}=1 $

(d) Calculate $ E\left[\mathbf{X}^{j}\mathbf{Y}^{k}\right] $ .

$ E\left[\mathbf{X}^{j}\mathbf{Y}^{k}\right]=\frac{\partial^{j+k}}{\partial\left(i\omega_{1}\right)^{j}\partial\left(i\omega_{2}\right)^{k}}\Phi_{\mathbf{XY}}\left(\omega_{1},\omega_{2}\right)|_{i\omega_{1}=i\omega_{2}=0}=\frac{\partial^{j+k}}{\partial\left(i\omega_{1}\right)^{j}\partial\left(i\omega_{2}\right)^{k}}\left[\left(1-i\omega_{1}\right)^{-1}\left(1-i\omega_{2}\right)^{-1}\right]|_{i\omega_{1}=i\omega_{2}=0} $$ =j!\left(1-i\omega_{1}\right)^{-\left(j+1\right)}k!\left(1-i\omega_{2}\right)^{-\left(k+1\right)}|_{i\omega_{1}=i\omega_{2}=0}=j!k! $

(e) Calculate the correlation coefficient $ r_{\mathbf{XY}} $ between $ \mathbf{X} $ and $ \mathbf{Y} $ .

$ r_{\mathbf{XY}}=\frac{Cov\left(\mathbf{X},\mathbf{Y}\right)}{\sigma_{\mathbf{X}}\sigma_{\mathbf{Y}}}=\frac{E\left[\mathbf{XY}\right]-E\left[\mathbf{X}\right]E\left[\mathbf{Y}\right]}{\sigma_{\mathbf{X}}\sigma_{\mathbf{Y}}}=\frac{1-1\cdot1}{\sigma_{\mathbf{X}}\sigma_{\mathbf{Y}}}=0. $