Let X1, X2, and X3 represent the times necessary to perform three successive repair tasks at a certain service facility. Suppose they are independent, normal rv's with expected values ?1, ?2, and ?3 and variances ?12, ?22, and ?32, respectively. (a) If ?1 = ?2 = ?3 = 70 and ?12 = ?22 = ?32 = 18, calculate P(To ≤ 228) and P(174 ≤ To ≤ 228). P(To ≤ 228)= P(174 ≤ To ≤ 228)= (b) Using the ?i's and ?i's given in part (a), calculate both P(64 ≤ X) and P(68 ≤ X ≤ 72). P(64 ≤ X)= P(68 ≤ X ≤ 72)= (c) Using the ?i's and ?i's given in part (a), calculate P(−12 ≤ X1 − 0.5X2 − 0.5X3 ≤ 6). P(−12 ≤ X1 − 0.5X2 − 0.5X3 ≤ 6) =
Continuous Probability Distributions
Probability distributions are of two types, which are continuous probability distributions and discrete probability distributions. A continuous probability distribution contains an infinite number of values. For example, if time is infinite: you could count from 0 to a trillion seconds, billion seconds, so on indefinitely. A discrete probability distribution consists of only a countable set of possible values.
Normal Distribution
Suppose we had to design a bathroom weighing scale, how would we decide what should be the range of the weighing machine? Would we take the highest recorded human weight in history and use that as the upper limit for our weighing scale? This may not be a great idea as the sensitivity of the scale would get reduced if the range is too large. At the same time, if we keep the upper limit too low, it may not be usable for a large percentage of the population!
Let X1, X2, and X3 represent the times necessary to perform three successive repair tasks at a certain service facility. Suppose they are independent, normal rv's with
- The quantity represents the
probability that the difference between X1 and the sum of X2 and X3 is between −12 and 6. - The quantity represents the probability that the difference between X1 and the average of X2 and X3 is between −12 and 6.
- The quantity represents the probability that the difference between X3 and the sum of X1 and X2 is between −12 and 6.
- The quantity represents the probability that the difference between X3 and the average of X1 and X2 is between −12 and 6.The quantity represents the probability that X1, X2, and X3 are all between −12 and 6.
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