Chances are because of not enough cooling. Also your PSU may cause this problem.
HD 7750 Requires a 450 W PSU whereas yours is only 400 W. As a result of this, When your PC is Idle, Graphic card requires less Power, but when it is under heavy load, all other parts like mobo, Processor, Graphic Card starts demanding a little more power for its fans and other things. As a result PSU is unable to manage that and Your system Crashes.
Also cooling could be the problem. If the cooling is not enough, The graphic card heats up faster and when it gets heated to a level, it stops displaying in order to prevents the parts from being damaged.
The chances of cooling problem is less as it is a new Graphic Card, like you said.
This is all I could think of. Your ram is not an issue. A system never crashes unless its ram gets damaged. Your system will become slow, but wont crash because of less ram. I have played Most Wanted 2012 in 2 GB RAM and it works fine. So no issue of ram.