Yes the media makes them seem like part of the Gov, but they are not.
Most of the Fed profits are handed over to the US Treasury, so it knows who the boss is, but Congress is treated with contempt by the Fed.
Looks correct:
The
Insanely Profitable Federal Reserve
http://www.thedailybeast.com/articles/2014/01/14/the-insanely-profitable-federal-reserve.html....
It also makes a metric ass-ton of money in the process, kicking this cash back upstairs to the U.S. government that permits it to operate.
How profitable is the Federal Reserve?
Victoria McGrane's story at the Wall Street Journal tells us about the Fed's incredibly lucrative 2013 (emphasis mine):
The Federal Reserve sent about $77.7 billion in profits to the Treasury Department in 2013.
The Fed in a statement released Friday said it made an estimated $79.5 billion in net interest income, a total largely driven by the $90.4 billion in interest income it made on its portfolio of Treasurys, mortgage bonds and other securities.