probability theory and statistics 13

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

  1. (a) Calculate E[Z], the expected value of the absolute difference in lifetimes of the two devices. Hints: (a) This is not E[X −Y] = E[X]−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.
  2. (b) Explain why your answer makes sense. Was there an easier way to do this problem?
  3. (c) Suppose X and Y have different parameters (λX and λY ). What is E[Z] in this case?
  1. 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.
    1. (a) What is the expected value of the time needed to produce 25 devices? What is the standard deviation of this quantity?
    2. (b) The technician can go home at the end of his 6-hour shift or when he’s completed 25 devices, whichever comes first. Use the central limit theorem to estimate the probability that he goes home early.
  2. 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 ≤ X ≤ 76 with probability 0.99?
  3. Use the Central Limit Theorem Approximation to answer the following:
    1. (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 “A” grade is 90% (that is, at least 45 correct), what is the probability of getting an “A”?
    2. (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 “A” grade is 90% (that is, at least 90 correct), what is the probability of getting an “A”? Why is this different than the answer to part (a)?
    3. (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 “A” grade is 90%, what is the probability of getting an “A”? Why is this exam easier (or harder) than the one in part (b)?
    4. (d) Repeat the calculations in (a) and (b) using the “continuity correction”. Does this make much of a difference in your results?
 
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