Abstract
In this paper, we study the discounted free Gerber–Shiu function for the compound binomial risk model with by-claims and randomized dividend policy. Specifically, explicit expression for the discounted free Gerber–Shiu function is obtained. This result allows us to derive formulae for some useful insurance quantities, including the ruin probability, the probability function of the deficit at ruin, the joint probability function of the surplus immediately before ruin and the deficit at ruin, and the probability function of the claim causing ruin.