This paper studies properties of the score distributions of calibrated log-likelihood-ratios that are used in automatic speaker recognition. We derive the essential condition for calibration that the log likelihood ratio of the log-likelihood-ratio is the log-likelihood-ratio. We then investigate what the consequence of this condition is to the probability density functions (PDFs) of the log-likelihood-ratio score. We show that if the PDF of the non-target distribution is Gaussian, then the PDF of the target distribution must be Gaussian as well. The means and variances of these two PDFs are interrelated, and determined completely by the discrimination performance of the recognizer characterized by the equal error rate. These relations allow for a new way of computing the offset and scaling parameters for linear calibration, and we derive closed-form expressions for these and show that for modern i-vector systems with PLDA scoring this leads to good calibration, comparable to traditional logistic regression, over a wide range of system performance.