## 1. The reaction time of a driver to visual stimulus is normally distributed with a mean of 0.2 seconds and a standard deviation of 0.1 seconds. 1‐1. (2 points) What is the probability that a reaction requires more than 0.5 seconds? 1‐2. (2 points) What is the probability that a reaction requires between 0.4 and 0.5 seconds? 1‐3. (2 points) What is the reaction time that is exceeded 95% of the time? 2. Spherical Uniform Distribution (Google! You do not have to explain why): 2‐1. (2 points) How can we pick a set of random points uniformly distributed on the unit circle x12 + x 2=1? 2‐2. (2 points) How can we pick a set of random points uniformly distributed on the 4‐dimensional unit 2 2 2 2 2 sphere x1 + x2 + x3 + x4 + x5 =1? 3. The random variable X has a binomial distribution with n = 19 and p = 0.4. Determine the following probabilities. (You may use computer. But, you have to show the formula.) 3‐1. (2 points) P(X ≤ 12) 3‐2. (2 points) P(X ≥ 18) 3‐3. (2 points) P(13 ≤ X < 15) 4. (2 points) Show the mean and the variance of the triangular distribution with lower limit a, upper limit b and mode c, where a < b and a ≤ c ≤ b. (You must show why.) 5. (2 points) An electronic office product contains 5000 electronic components. Assume that the probability that each component operates without failure during the useful life of the product is 0.999, and assume that the components fail independently. Approximate the probability that 10 or more of the original 5000 components fail during the useful life of the product. 6. Consider the following system made up of functional components in parallel and series. C2 0.80 C1 0.90 C4 0.95 C3 0.85 6‐1. (2 points) What is the probability that the system operates? 6‐2. (2 points) What is the probability that the system fails due to the components in series? Assume parallel components do not fail. 6‐3. (2 points) What is the probability that the system fails due to the components in parallel? Assume series components do not fail. 6‐4. (2 points) Compute and compare the probabilities that the system fails when the probability that component C1 functions is improved to a value of 0.95 and when the probability that component C2 functions is improved to a value of 0.85. Which improvement increases the system reliability more? 7. (2 points) Suppose that the joint distribution of X and Y has probability density function f(x, y) = 0.25xy for 0 < x < 2 and 0 < y < 2. Compute V(2X + 3Y). (Show all your work.)

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