Statistics

2. 66 students took the MATH/STAT 414 midterm (independently due to random seats), and the amount of time (in minutes) needed by any student to finish the exam was Exponentially distributed, i.e., T; ~ Expo(1) for j = 1, 2, ...,66, where T, is the time needed for the j-th student to finish the exam. Tak believes that the average time needed for each student was 60 minutes. The exam started at 10:35am. (a) (2 pts) What is the probability that at least one of the students finished the exam before 11:35am (i.e., in an hour)? Hint: The event that "no students will finish the exam before 11:35am" can be formulated as {T1 > 60, T2 > 60, . .., 766 > 60}. After answering this, use the independence of events. (b) (5 pts) Surprisingly, Tak observed that one student, whose name will remain hidden for his or her own safety, finished the exam first at 11:05am (in 30 minutes!). Without loss of gener- ality, let us assume that this student was the 66th student, i.e., j = 66 for this student. Given this information (i.e., after 30 minutes), what is the probability that at least one more student (among the remaining students) finished the exam before 11:35am? Hint: You may want to find the following probability to compute the probability of interest: P(T1 > 60, T2 > 60, . .., T65 > 60 | 71 > 30, T2 > 30, ..., T65 > 30). For example, {71 > 30} is in the condition because when the 66th student finished the exam at 11:05am, this student (7 = 1) did not finish the exam