Lily was a homeless girl that ran from home after years of abuse from her dad (he is currently in prison).

Josh saw Lily sitting alone in a sidewalk. Lily was mute and wrote to Josh with napkins and a spare marker she had lying around. Writing her name and requested to have her live with him. At first, Josh was hesitant since he doesn't know her, however the pity outweighed his thoughts of declining the request. And thus he let her live in his apartment, and the rest is history!