# probability theory and statistics 13

1. *X *and *Y *are exponential random variables, with the same Î» parameter; these represent the lifetimes of two identical devices. Let *Z *= |*X *âˆ’ *Y *|, the absolute difference of *X *and *Y *.

- (a) Calculate
*E*[*Z*], the expected value of the absolute difference in lifetimes of the two devices.*Hints: (a) This is*not*E*[*X*âˆ’*Y*] =*E*[*X*]âˆ’*E*[*Y*] = 0*. (b) Use the Law of Total Probability to split the problem into two events; one where X fails first, and one where Y fails first.* - (b) Explain why your answer makes sense. Was there an easier way to do this problem?
- (c) Suppose
*X*and*Y*have different parameters (Î»*X*and Î»*Y*). What is*E*[*Z*] in this case?

- A technician assembles electronic devices. The mean time required to assemble a device is 15.0 minutes, with a standard deviation of 3.0 minutes. The devices are all the same, and the time required to assemble one is independent of the time to assemble any other.
- (a) What is the expected value of the time needed to produce 25 devices? What is the standard deviation of this quantity?
- (b) The technician can go home at the end of his 6-hour shift or when heâ€™s completed 25 devices, whichever comes first. Use the central limit theorem to estimate the probability that he goes home early.

- Let
*X*be a random variable with mean 75 and standard deviation 15. Suppose we make multiple measurements of*X*, such that*X*1,*X*2… are an i.i.d. sequence. (a) Assuming that the sample mean*X*is approximately normally distributed, how many measurements*n*will be required to ensure that 74 â‰¤*X*â‰¤ 76 with probability 0.99? - Use the Central Limit Theorem Approximation to answer the following:
- (a) A multiple-choice exam has 50 questions. Suppose the probability of getting each question correct is
*p*= 0.8, independent of the other questions. If the cutoff for an â€œAâ€ grade is 90% (that is, at least 45 correct), what is the probability of getting an â€œAâ€? - (b) A multiple-choice exam has 100 questions. Suppose the probability of getting each question correct is
*p*= 0.8, independent of the other questions. If the cutoff for an â€œAâ€ grade is 90% (that is, at least 90 correct), what is the probability of getting an â€œAâ€? Why is this different than the answer to part (a)? - (c) A multiple-choice exam has 100 questions. 50 of the questions have
*p*= 0.7, and the remaining 50 questions have*p*= 0.90. If the cutoff for an â€œAâ€ grade is 90%, what is the probability of getting an â€œAâ€? Why is this exam easier (or harder) than the one in part (b)? - (d) Repeat the calculations in (a) and (b) using the â€œcontinuity correctionâ€. Does this make much of a difference in your results?

- (a) A multiple-choice exam has 50 questions. Suppose the probability of getting each question correct is