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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 X1, X2… 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?