30 - Day Notice with Prepaid Last Months Rent

Hey guys, I've got a quick question which has thrown me for a bit of a spin. I am finally closing escrow on my first duplex, which will be owner occupied, so the time has come to put in the notice at my current residence. I have done a lot of researching and talking to people on this matter, but I am still not 100% sure, so here it goes.

I am currently on a month-to-month lease where i have paid my last months rent along with a security deposit. Now according to what I am reading on the California Tenant-Landlord Laws, in my situation I obviously am only obligated to giving a 30 day notice. The law states I can give my notice any time I want, whether it be the 1st of the month, the 5th, the 12th, it doesn't matter. But here is where I am a bit confused. When I have already paid my full last months rent does this mean that I am entitled to stay and pay that last months rent, or is the last month pro-rated and refunded depending on when the 30 days ends?

In regards to my current lease, all it outlines that I have paid first, last, and deposit and that we are required to give a 30 day notice, but nothing more. 

Thanks in advice for all your knowledge and help!