Bayesian density functions for Gaussian pulse shapes in Gaussian noise

The probability density function for the amplitude of a Gaussian-shaped pulse of unknown arrival time is derived together with its mean and variance, when the time argument of the pulse is randomly distributed according to a Gaussian probability distribution or a uniform distribution. Then using the amplitude density as a prior distribution, the marginal density function for the pulse plus stationary Gaussian noise and its attendant expression for the probability of detection are derived. Applications to the calculation of detection probabilities in the presence of pulse peak location errors are cited.