Note 4 at Mr 10:50: Bartimaeus cast away his outer garment, which would be comparable to a coat that we would wear. This was done apparently because of the haste with which he was making to go to Jesus, and he didn't want anything to cumber him or get in his way.

Different classes of society could also be recognized by their clothing, as can be seen by the Pharisees (Mt 23:5). Some have suggested that beggars could also be distinguished by their outer garments, and therefore this could have been an action on Bartimaeus' part that showed his unwavering faith that Jesus would heal him. He wouldn't be needing his beggar clothes anymore.