(b) (6%) The third question has diffieulty level 3 while the fourth question has difficulty level 5. Denote U as the random variable rep presenting the difference in time to come up with the correct answers for the two questions (time for the fourth minus time for the third). Write down the PDF of U.

題組內容

11. (30%) A study show that in an exan the time required to come up with the correct answer for each question is a cont tinuous memoryless random variable with the ex expected value equal to the difficulty level of the question. The time to answer each question is independent of each other.

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.

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