What was found
APPROX 15 MIN INTO FLT RECEIVED AN E2 IMP BYPASS MESSAGE. RAN QRH AND CONTINUED. MESSAGE WENT OUT VERY SHORTLY AFTER IT APPEARED. APPROX 35 MIN INTO FLT, ITT JUMP FROM GREEN INDICATION TO RED AND BACK TO AMBER AND GREEN. PULLED NR 2 THRUST LEVER BACK. ITT WENT BACK INTO GREEN. ADVANCED THE THRUST LEVER AND ITT SHOT RIGHT BACK INTO RED AND ENGINE SHUT ITSELF DOWN. RAN QRH ENGINE FAILURE CHECKLIST, PRECAUTIONARY ENGINE SHUTDOWN CHECKLIST, EMERGENCY DESCENT CHECKLIST. NOTIFIED ATC AND GOT CLEARANCE FOR EMERGENCY DESCENT. F/A REPORTED SMOKE IN CABIN. DECLARED EMERGENCY. FIRE TRUCKS ROLLED OUT BEHIND & CLEARED AIRCRAFT OF ANY SMOKE COMING FROM NR 2 ENGINE. MX REMOVED AND REPLACED NR 2 ENGINE.