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The stochastic process at the output of a matched filter, when the latter is excited by its proper signal in additive white noise, has a mean function proportional to its covariance function. Sample path properties of a Gaussian process with the mean proportional to the covariance, conditioned such that it assumes a given value at the instant of the peak in the mean, are independent of signal amplitude. Formal and rigorous proofs and a detection-theoretical interpretation of this result are presented. It is then applied to the calculation of the detection probability of a rectangular signal of unknown time of arrival and to bounding the threshold effect in the estimation of the time of arrival. A novel passage time result is derived in the Appendix.