We consider the origin of noise and distortions in power spectral estimates of randomly sampled data, specifically velocity data measured with a burst-mode laser Doppler anemometer. The analysis guides us to new ways of reducing noise and removing spectral bias, e.g. distortions caused by modifications of the ideal Poisson sample rate caused by dead time effects and correlations between velocity and sample rate. The noise and dead time effects for finite records are shown to tend to previous results for infinite time records and ensemble averages. For finite records we show that the measured sampling function can be used to correct the spectra for noise and dead time effects by a deconvolution process. We also describe a novel version of a power spectral estimator based on a fast slotted autocovariance algorithm.