May I get help please?

12. Suppose that a certain system contains three components that function inde- pendently of each other and are connected in series, so that the system fails as soon as one of the components fails. Suppose that the length of life of the first component X1 ~ exp(8 = 001), measured in hours; the length of life of the second component X2 ~ exp(8 = 003), measured in hours; and the length of life of the third component Xy ~ exp(# = .006), measured in hours. Hint: Use results and concepts taught in class. Refer to class notes! (a) Suppose that the random variables X1, X2, Xs are independent and X, ~ exp( 8,) for i = 1,2,3. Let Y, = min(X1, X2, X). Show that Y, ~ exp(81+ (b) Determine the probability that the system will not fail before 100 hours. 13. Suppose that a certain examination is to be taken by five students independently of one another, and the number of minutes required by any particular student to complete the examination has an exponential distribution with a mean time of 40 minutes. Suppose that examination begins at 9:30am. Hint: Use results and concepts taught in class. Refer to class notes! (a) Determine the probability that at least one of the students will complete the examination before 10:15am. (b) Determine the probability that no two students will complete the examina- tion within 10 minutes of each other. (Hint: Recall the light bulb example where we determine the distribution of the length of time until one of the n bulbs fails.)