Physical meaning of average values of random signals
Dilip: I'm confident that what you said is correct but the weirdo sample path that you refer to ( the one with probability zero ) can be used to estimate the autocorrelation at different lags, namely $\hat{R}_{xx}(\tau)$. This is true even though it's only one realization and this is the case only because of the ergodicity assumption. So, to a person coming in new, my opinion is that your answer doesn't give enough significance to the one realization. It's a weirdo but a useful weirdo for sure  !!!!!