What was found
(CAN) ON CLIMB, BURN ODOR, SMOKE WHERE DETECTED IN COCKPIT. PILOT ADVISED ATC, DIVERTED TO ALTERNATE AIRPORT WHICH HAD VFR CONDITIONS AT TIME. WHILE FLYING INTO HEAVY RAIN, TURBULENT AIR, RT ENG FIRE WARNING LIGHT ILLUMINATED. RT ENG WAS IMMEDIATELY SHUTDOWN. AC LANDED WITHOUT INCIDENT. INSP OF RT ENG DID NOT SHOW ANY SIGNS OF FIRE, SMOKE, OR BLEED AIR LEAK. FIRE DET SYS WAS TESTED USING AC FIRE DETECTOR TEST SWITCH. FIRST TIME, RT NR2 FIRE DET INDICATED U/S. MALF COULD NOT BE DUPLICATED. BURN ODOR AND SMOKE WAS FOUND TO BE COMING FROM DISTANCE MEASURING EQUIPMENT (DME) DISPLAY SECTION OF HORIZ SITUATION INDICATOR (HSI) CIRCUIT BOARDS. DME CB WAS PULLED, GROUND RUN CARRIED OUT, AC FERRIED FOR REPAIR.