I will devloping a game using the XNA Studio. The game will not be comercial this is just a freeware game that people will be able to download from the internet. if the game will be freeware do I need to pay money?
Well I suggest you have a look at the XNA Express using C#. You will need to pay $99AUD (Don't know the USD price) to get access to loading the game onto your Xbox. However you can't sell it at all, I believe you can put the source up so that other XNA peepz can play your game though.
If you want to make an actual arcade game you need the real SDK which costs a wee bit of money and comes preloaded on an apple powermac :D.
EDIT:
I stand pleasantly corrected by @JohnRudy. It appears microsoft decided that people might want to get compensation or at least some playtime out of their arcade games. I'm glad they've removed the draconian restrictions on getting your game into the arcade.
So bottom line is: for Xbox 360, no you cannot create a freeware game that is distributed via XBL. You can release the source and others can download and run it on the Xbox, but it will cost both you and the end user money since you would both need a Creators Club membership.
For the PC though you are free to develop and distribute as you wish.