Measuring X-Ray Variability in Faint/Sparsely Sampled Active Galactic Nuclei

We study the statistical properties of the normalized excess variance of variability process characterized by a "red-noise" power spectral density (PSD), as in the case of active galactic nuclei (AGNs). We perform Monte Carlo simulations of light curves, assuming both a continuous and a sparse sampling pattern and various signal-to-noise ratios (S/Ns). We show that the normalized excess variance is a biased estimate of the variance even in the case of continuously sampled light curves. The bias depends on the PSD slope and on the sampling pattern, but not on the S/N. We provide a simple formula to account for the bias, which yields unbiased estimates with an accuracy better than 15%. We show that the normalized excess variance estimates based on single light curves (especially for sparse sampling and S/N < 3) are highly uncertain (even if corrected for bias) and we propose instead the use of an "ensemble estimate," based on multiple light curves of the same object, or on the use of light curves of many objects. These estimates have symmetric distributions, known errors, and can also be corrected for biases. We use our results to estimate the ability to measure the intrinsic source variability in current data, and show that they could also be useful in the planning of the observing strategy of future surveys such as those provided by X-ray missions studying distant and/or faint AGN populations and, more in general, in the estimation of the variability amplitude of sources that will result from future surveys such as Pan-STARRS and LSST.