On the ratio of two random variables

The probability P(k) that two non-negative random variables X and Y are within K ≡ 20 log10k dB of each other is investigated. When the joint probability density function is symmetric in X and Y, then P(k) = 2FZ(k) - 1, where FZis the cumulative distribution function of the ratio Z ≡ (X/Y). Finally, P(k) is evaluated for the case in which X and Y are correlated Rayleigh variates.