申論題

13. (8%) Alice and Bob are playing a gane against the house (host of the game). A circle is drawn on the wooden wall, with the inner region (disk) assigned to Alice and the outer region assigned to Bob. The center is labeled as the origin (0, 0) and Alice and Bob take turns to throw darts to the c circle. When Alice throws a dart the co e coordinates of the landing point are a Gaussian pair of independent randomn variables each with zero mean and variance 1. When Bob throws the dart the coordinates are also a Gaussian independent pair but with zero mean and variance 4. Whenever Ali Alice's dart lands on Bob's region, she must pay $2 dollar to the house； whenever Bob's dart lands on Alice's region, he must pay S1 dollar to the house. Find the disk radius that minimizes the players' average expense.