The pilot was on the last leg of his first trip to Cairns. He was cleared by Cairns Approach to descend to 3000 ft, not below the DME (Distance Measuring Equipment) steps. Subsequently the Approach Controller observed, on the radar, that the aircraft had descended to 6000 ft at 27 DME. The minimum altitude when conducting a DME Arrival at this distance is 6500 ft. The aircraft was instructed to climb to 6500 ft and the flight continued normally to Cairns. The pilot advised that he was not expecting to be given a DME Arrival and had not checked the procedure prior to the flight. However, he checked the procedure when cleared by Cairns and had his wife hold the approach plate for him. About that time his son became ill and his wife dropped the plate. By the time he had recovered the situation the aircraft had been allowed to descend below the profile. The aircraft was in cloud at the time and the pilot reported that it was turbulent.