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I am looking forward for the badge - Beta, How i can actively participate in Beta, I don't know what do they mean by it.
I can't see any link about it.
If you have not participated in the private beta of a site, you cannot earn the badge.
For Stack Overflow, the private beta ended almost 4 years ago.
This badge is similar to the precognitive badge (Followed the Area 51 proposal for this site before it entered the commitment phase.), which will never be earned on sites such as Stack Overflow that predate Area 51.
To answer your questions as they pertain to sites currently on Area 51 - after a site gets accepted, it will go through a private beta phase. If you are part of that phase and participate enough to earn the beta badge, it is yours. Badges are taken away only if we confirm they were earned through fraud (say using sock puppets, voting rings etc) - so even if your participation levels drop during the private beta, you will retain the badge if you earned it.
The Beta badge is for people who were with any new StackExchange site when it first started. It is not possible to earn this badge if you were not an early adopter.