cut floor joist not pocketed into brick wall

Hi All, thanks in advance for the advice.
We are doing a mostly-gut remodel of a small townhouse. We recently removed a wall and opened the ceiling above and found that one of the floor joists for the second floor had been cut about 4-5 feet from the wall. There are some wood scraps inserted into the gap, maybe to hold the wiring up?
The other floor joists all extend from one side of the room to the other, and are pocketed into the brick walls on both sides. I'd like to replace the missing section of joist and sister it to another board, but hesitating because I'm confused by the fact that there is no existing pocket for the end of the missing section of joist to slide into. Either there was never a pocket and this floor joist was just floating in space, or someone did a very convincing brick repair where the pocket used to be, or...well, that's all I've got.
Do any of you pros know of a good reason why I shouldn't cut a pocket in that wall to insert the ends of the new boards? Does this seem like a reasonable solution? (That ductwork will be coming out as well.)

 

Not a fan of pocketed joists as the ends have a tendency to rot over time, but you've got what you've got. Rather than cut a new pocket to fix that joist, I'd run a doubled up 2 x 8? between the two joists on either side, close to the wall, and use a joist hanger to support the end of the repaired joist.
Would certainly be better support than what you have.

 

CarbideTipped Thanks for the reply! I've never owned a brick-wall home before, so I hadn't considered that the joist ends are probably just being repeatedly wetted. It does look like some of the joists in that ceiling have been replaced - maybe that's why. I like the idea of using the joist hanger instead of cutting the brick, thanks!