Fibre sizes have to be described by both diameter d and length l. There is a stochastic relationship between D and L which may be given by the bivariate distribution function for the bivariate stochastic variable (D, L). A measure w = ad^rl^s is introduced and formulae are developed for the distribution function, the expected value and the variance for the random variable W = aD^rL^s. The expression for w was chosen because such important characteristics as fibre surface area (approximately), volume, aspect ratio and settling speed are covered. Corresponding expressions are given for (D_w, L_w), i.e. the diameter and the length of fibres sampled randomly from a fibre population with a probability proportional to the w-value of the individual fibres. By combination the random variable X_w = kD_w^pL_w^q, is described. When (D, L) is assumed to be bivariate log-normally distributed the derived distributions will also be log-normal, respectively, bivariate log-normal and expressions are given for the parameters of these distributions in terms of the parameters of the distribution of (D, L), i.e. the mean value and the variance of the marginal distribution of ln D, respectively, In L and the correlation between ln D and ln L.