TV Hums when TV is on and also shuts off by itself
Definitely the inverter! Take it apart (Mostly just screws) and if you or your friend can solder, then just re-solder all of the transformers and anything that looks sloppy. This is the best case scenario fix... otherwise if that doesn't work or is not an option then just get the Model number off of it and buy a used inverter on e-Bay. Just make sure you follow the sellers requirements for installation.
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That would be the inverter, The inverter on my poloroid LCD is very noisy when the TV is off, Be it's quite normal and isn't much to worry about for now.
The switching off however is not so normal, I would suggest removing the back cover and blowing out any dust and dirt with compressed air, Even just aiming a desk fan at the vents on the back of the TV should help greatly. My uncles LCD switches off every hour so they placed a fan underneath it (It's wall mounted) aimed up to the base of the tv and it's worked fine ever since.
The TV may be getting too hot, possibly use a computer fan on the vents and dust it off every now and then.