Am I sinful or did I cause the death of my infant brother after I discharged him from the hospital at my own responsibility despite the doctor's warning, and then he died a few hours after his discharge?

It is not permissible to discharge the child from the hospital after the doctor has warned of the danger involved. The sin falls upon both the father and the doctor. The guarantee (liability) rests with the father, as he directly initiated the child's discharge. He must offer expiation, which is: freeing a believing slave; if he cannot find one, then fasting for two consecutive months, along with repentance. Additionally, he and his 'aqilah (male relatives on the father's side) owe the diyah (blood money), unless the heirs forgive it. Furthermore, whoever is liable for the guarantee does not inherit from it if he is an heir.