How can unauthorized movie purchases be made on my 1st generation Apple TV?
I know this is a month old, but check your ir remote. My Fios remote is programmed to control my Apple TV. If the Apple TV is on, the remote may be triggering the purchases when you use the up, down and menu keys on it. Since you are watching another show through a different input, you won't even notice it until you get the bill. 
Reprogram the universal remote or tuck the Apple TV where the IR signal can't SE it.