To celebrate a big achievement of mine, I invited friends to my house for a party. Now, I am thinking how many bottles of beer I need to order. Each bottle of beer costs me $2. If there is any leftover beer, I will drink it at later times. In this case, each bottle of the beer is worth $1 to me. When my friends come to the party, enjoy the party and drink the beer, I feel happy, too, and it gives me joy that is worth $5. If any of my friends cannot drink enough because of the supply shortage, I feel sad, and it is worth 0.5 dollar cost per bottle. The quantity of beer needed at the party follows a uniform distribution between 100 and 200 bottles. (You may assume the demand is a continuous uniform distribution.) Determine how many bottles of beer I need to buy to minimize my expected cost (or maximize my profit or happiness).

To celebrate a big achievement of mine, I invited friends to my house for a party. Now, I am thinking how many bottles of beer I need to order. Each bottle of beer costs me $2. If there is any leftover beer, I will drink it at later times. In this case, each bottle of the beer is worth $1 to me. When my friends come to the party, enjoy the party and drink the beer, I feel happy, too, and it gives me joy that is worth $5. If any of my friends cannot drink enough because of the supply shortage, I feel sad, and it is worth 0.5 dollar cost per bottle. The quantity of beer needed at the party follows a uniform distribution between 100 and 200 bottles. (You may assume the demand is a continuous uniform distribution.) Determine how many bottles of beer I need to buy to minimize my expected cost (or maximize my profit or happiness).