The amplitude of a sonar echo from a fish depends upon the species and size of the fish, acoustic wavelength, aspect, position of the fish in the sonar beam, range and backscattering cross section. We simplify the problem to a single species and size of fish, vertically downward echo sounding, single aspect, and nonoverlapping echoes. After removal of attenuation due to range and absorption two random functions remain. The position of the fish in the sonar beam is random and the scattering cross section for each trail is random. We assume that the fish have a uniform density (number/m3) and calculate the probability density function (PDF) for insonification and reception. We assume that the PDF of the envelope of the echo (excluding the variability of insonification and reception) has a Rayleigh PDF. Assuming two PDF’s are independent, we calculate the PDF of the echo envelopeswE(e).wE(e) depends upon the beamwidth of the sonar and the mean backscattering cross section. The theoretical PDF has the same shape as the measured PDF of echoes from alewife in Lake Michigan. We use the fit of the PDF’s to estimate the backscattering cross section and fish density. This calibrates the echo‐integration processing system. A profile of the density of alewife in Lake Michigan is shown.Subject Classification: [43]30.40; [43]80.40.