When you are hoping for something, in this case for someone to have a good day. You are desiring things to happen, you set an expectatation that YOU WANT their day to go well. When you say I BELIEVE your day is going well. You then accept it as fact that they are having a good day because you think they are having a good day, regardless of the day that person may be having. So you would say I hope you have a good day and smile. Or I believe your having a good day and smirk.

They are not the same." I hope.. " is a wish. You are wishing the person a great day. The second "I believe" means that you are assuming you already KNOW that the person is having a great day. You would usually not use this expression if you intend to be polite and wish the person a great day.