Communication, Networking, Signal and Image Processing (CS)

Question 1: Probability and Random Processes

January 2004

**3. (30 pts.)**

Let $ \mathbf{X}\left(t\right) $ be a real continuous-time Gaussian random process. Show that its probabilistic behavior is completely characterized by its mean $ \mu_{\mathbf{X}}\left(t\right)=E\left[\mathbf{X}\left(t\right)\right] $ and its autocorrelation function $ R_{\mathbf{XX}}\left(t_{1},t_{2}\right)=E\left[\mathbf{X}\left(t_{1}\right)\mathbf{X}\left(t_{2}\right)\right]. $