Linking Autocorrelation of Gaussian Random Process to Conditional Probabilities

For a zero-mean Gaussian random process , the probability density function (pdf) of the conditional random variable is Gaussian. Its properties can be calculated from the autocorrelation (or power spectral density) of the process, (or ).
The conditional expected value is and the conditional variance is given by
The probability is calculated in Wolfram Language as
The connection between the autocorrelation function and the probability is highlighted here. Three autocorrelation functions are considered: , and . The autocorrelation function and associated power spectral density are plotted. The correlation time for each autocorrelation function is shown, where .
is shown as a function of the threshold and lag . This plot shows that approaches as lag increases beyond the correlation time. This behavior is expected since the random variables and are approximately uncorrelated and, for multivariate Gaussian random variables, uncorrelated implies statistical independence.
The conditional pdf is plotted and the conditional expected value, conditional variance and shown; is shown as the shaded area. The conditional pdf and the shaded area change as the lag , threshold and conditional variable are varied.


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[1] K. S. Shanmugan and A. M. Breiphol, Random Signals: Detection, Estimation and Data Analysis, New York: Wiley, 1988.
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