Light bulbs produced by rhett's lighting and by baldwin co. are of the same price but different quality. those from rhett's have lifetimes with the exponential(2) distribution and those from baldwin have lifetimes with the exponential(3) distribution. all light bulb lifetimes are mutually independent. this problem has a mix of computations to make by hand and by computer. (a) (by hand.) suppose you buy one bulb of each brand and use them until they fail. what is the probability that the rhett's light bulb has the longer life? (b) (by computer.) use pseudorandom simulation (e. g., with matlab or python) to esti- mate the probability in part (a), and compare the estimate to your answer from part (a). specifically, write code to produce pseudorandom pairs of light bulb lifetimes xi: lifetime of rhett's bulbi yi: lifetime of baldwin bulb i for i = 1, 2, n. the probability you are trying to estimate is approximately the fraction of values of i for which x; > yį. later in the semester, we will study guarantees associated with particular values of n. for now, experiment to convince yourself that increasing n lowers the variance of your estimates and that for large enough n, you get arbitrarily close to your answer in (a). (c) (by hand.) suppose you buy one bulb of each brand and start using them simultaneously. as soon as one fails, you conclude that the failing brand is the worse one and buy 98 more of the other one (the brand you infer to be better). what is the expected total life from your 100 bulbs? (d) (by computer.) instead of making your decision after one bulb failure, you could wait to decide which is the better brand after you have more information. suppose you buy k of each brand, and after all 2k of these have failed, you use the total life provided by each brand to infer which is the better brand. you then buy 100 – 2k of the brand that you infer to be better. what is the expected total life from your 100 bulbs? show this as a plot as a function of k and provide some qualitative interpretation for what you see. (e) (interpretation without new computations.) in terms of total expected lifetime of the bulbs you buy in part (d), what is the best choice of k?