Can I backup my purchased android app I need this information because,  if my data gets lost because of some reason I need to reinstall all my apps, but what about those which I've purchased.  Is it necessary to purchase those purchased apps again or else their is a way for backup.
You can backup any app you like, even purchased ones.
For your purchased apps: Make sure to log in to the Google Account you purchased it with. Else you might get "unlicensed" errors.