Yes, you can do it over your home cable connection. Also you may want to check if your ISP allows you to do so as most will not.
To do this you will need a static IP address assigned to you by your ISP, then you need to update the nameserver record with you domain registrar. Once this is done you can use a software like FoxServ and install it on your computer.
(FoxServ is an Apache / mySQL / PHP installer package for Windows and Linux. )
Also by doing this you are opening yourself to security risks, so be sure to secure your system.