RAM sticks are often consider as prime culprit behind random and creepy system failures. There exist a number third party fancy memory testing tools with extensive list of features. However a inbuilt ready to use RAM diagnostic tool also exist in all Windows PC, which is simple and yet powerful enough to do the memory test.
After completion of test it will automatically restart windows and after awhile it will show a notification balloon in task bar showing the status of test, if no error is found then the notification balloon would say ‘No errors were detected’
So, if RAM is fine then you need to look into other hardware or software problem source.