OK, so I just started using twitter and I'm confused. I have no interest in Justin Beiber, nor does anyone else I follow on Twitter. So why does my home page show Justin Beiber tweets? Or more specifically, why do I see so many tweets from people I'm not following, and none from the people I am following? And the majority of the random tweets are from subjects I'm not even interested in.
You may be seeing this as a result of an experiment in user engagement and retention by Twitter, where you see favourited tweets show up as if they were normal tweets and retweets.
This should explain why you're seeing tweets from accounts that seemingly have no relation to those you follow since they didn't explicitly retweet them.
You could also be seeing Promoted Tweets, but these will usually be marked as such and show a button where you can "dismiss" it from your timeline. With promoted tweets you can be targeted "against" one of your interests.