(e) (6%) The fifth question has diffculty level 7 while the sixth question has difficulty level 8. 'Due to time constraint, you have time to write only one question. What is the probability that the time to finish the fifth question is shorter than the time to finish the sixth question?

(a) (6%) The first and second questions both have dificulty level 2. Denote T as the random variable representing the total I time I required to come up with the correct answers for both questions. Write down the PDF of T.

(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.

(d) (6%) In another exarn, there are so many questions s all of ther. Fortunately, all questions in the exam have the same diffculty level 2. Denote s such that it seer ers impossible to finish V as the number of questions answered corrcctly for the exam within the total allowable time of 100 (time units). Write down the PMF of V.

(e) (6%) 50 students participate in an exam and all of them start with the first question at the same tine. The diffculty level of the first question is 2. What is the probability that the question is answered correctly within 2 time units by any of the 50 students?

(a) (6%) Let N be the number of functional sensors in any sub-region of area A. Wite down the PMF of N.

(b) (6%) Define the core of the distribution region as a sq quare with side length 2(D - R) centered at (0,0), where no bound dary effect of sensot distribution needs to be considered. If a wildfire occurs within the core, find the probability that it is successfully detected.

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.

(b) (6%), (7%) Please give their (Equations (2) and (3)) matching engineering (or physical) examples with discussion, respectively.

(a) (6%), (6%) Please first prove Equations (2) and (3), respectively

(d) (5%) Can one obtain identical Fourier series of any function g(t) by using 2T and T in Equation (1)? Explain your reasons within 30 words.

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