The axial transmission technique can measure the longitudinal wave velocity of an immersed solid. An elementary model of the technique is developed with a set of source and receivers placed in a semi-infinite fluid coupled at a plane interface with a semi-infinite solid. The acoustic fluid is homogeneous. The solid is homogeneous, isotropic, and linearly elastic. The work is focused on the prediction of the measured velocity (apparent velocity) when the solid is considered to have random material properties. The probability density functions of the random variables modeling each mechanical parameter of the solid are derived following the maximum entropy principle. Specific attention is paid to the modeling of Poisson's ratio so that the second-order moments of the velocities remain finite. The stochastic solver is based on a Monte Carlo numerical simulation and uses an exact semianalytic expression of the acoustic response derived with the Cagniard-de Hoop method. Results are presented for a solid with the material properties of cortical bone. The estimated mean values and confidence regions of the apparent velocity are presented for various dispersion levels of the random parameters. A sensibility analysis with respect to the source and receivers locations is presented.