Computer randomly shuts itself down.

Even when your computer may seem relatively cool when starting up, accumulation of dust and other particles blocking your fans and vents can heat up your computer extremely fast, causing it to shut down as a safety measure. Make sure your fans are working properly, and aren't blocked as well as any vents. The heat needs to be able to escape to function properly. Also make sure your computer is in a location that is ventilated nicely. If it is cluttered up in a tight location that can attribute to overheating as well. Basically it's mostly as WandererWillow described. Compressed air can be your best friend.