Yep this is normal. The system only compares sensor values from time to time. However, if there is a catastrophic failure it would trigger immediately.
BTW, where are your stock wheels/tires stored? If they are near your car when you park, the system will probably reset each time...which means you might not have a warning light at the first of each start-up.
As for price? It has come way down in the last year or so. OE sensors can be purchased from ebay for less than $100. Pacific brand # PMV-107J is what you want. Some tires shops will program for free. A piece of electrical for the light works too.