I broke my distal radius last week they put me in a full arm cast and they said they will make the cast smaller in a week will i get a below elbow cast? displaced btw?

As you improve/heal after a fracture, you need less and less plaster to protect the break. In a full arm cast you lose substantial function; with a below elbow cast you will be much more mobile and your recovery will be easier. They will not progress unless it is deemed safe for you.

The long arm cast prevents pronation and supination reducing forces that could distort the fracture after 2-3 weeks the fracture is often stable enough to reduce the cast to below the elbow.