There is only mention of tributes from Districts 1 through 12. Are children that live in the Capitol excluded from the Hunger Games?
Yes - children that live in the Capital are excluded. :)
That's actually the main point of them; Capital was revolted against, so these games are to remind them what kind of cost they pay for revolt.
From the first book:
The Treaty of Treason gave us the new laws to guarantee peace and, as our yearly reminder that the Dark Days must never be repeated, it gave us the Hunger Games. The rules of the Hunger Games are simple. In punishment for the uprising, each of the twelve districts must provide one girl and one boy, called tributes, to participate. The twentyfour tributes will be imprisoned in a vast outdoor arena that could hold anything from a burning desert to a frozen wasteland. Over a period of several weeks, the competitors must fight to the death. The last tribute standing wins. Taking the kids from our districts, forcing them to kill one another while we watch—this is the Capitol’s way of reminding us how totally we are at their mercy. How little chance we would stand of surviving another rebellion.
It's kind of meant as a constant reminder that "Capital rules; the last revolt, 75 years ago, is STILL being paid for.. Do you really want to try to dispute our rule again?"
The children from Capitol were excluded, because the games were meant to remind the defeated districts of the power of Capitol.
Spoiler alert: At the end of the third book, the punishment for the defeated Capitol is that the children of the leadership would be forced to participate in the final Hunger Games.
So, an alternative has been placed on the table. Since my colleagues and I can come to no consensus, it has been agreed we will let the victors decide ... What has been proposed is that in lieu of eliminating the entire Capitol population, we have a final, symbolic Hunger games, using the children directly related to those that held the most power.
Katniss is the deciding vote in favor