The daily flow rate of contaminants from an industrial plant is modeled by a normal random variable with a mean value of 10 units/day and a coefficient of variation of 20%. When the contaminant flow rate (CFR) exceeds 14 units on a given day, it is considered excessive. Assume that CFR between any two days is statistically independent. 1.) What is the probability that the industrial plant has an excessive CFR on a given day? 2.) Assume that the government agency in charge of the monitoring of the plant requires the measurement of CFR for 2 days. Moreover, based on existing implementing rules and regulations, any industrial plant will be charged with a violation if an excessive CFR is observed during the 2-day period. What is the probability that the plant will NOT be charged with a violation? 3.) Suppose that the industrial plant decides to limit the probability of violating the 2-day regulation to 1%. What should be the daily mean CFR? Assume that the plant can reduce the mean daily CFR by improving its own in-house chemical process. 4.) Suppose that there is a new proposal in 2022 to change the regulation such that CFR will be measured for 4 days and that any manufacturing plant will be charged with a violation if the excessive CFR is observed in MORE THAN ONE of the 4 days. Which scheme is better (i.e., less strict) for the manufacturing plant? the 2-day scheme or the 4-day scheme? Justify your answer. Thank you
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!
The daily flow rate of contaminants from an industrial plant is modeled by a normal random variable with a
1.) What is the
2.) Assume that the government agency in charge of the monitoring of the plant requires the measurement of CFR for 2 days. Moreover, based on existing implementing rules and regulations, any industrial plant will be charged with a violation if an excessive CFR is observed during the 2-day period. What is the probability that the plant will NOT be charged with a violation?
3.) Suppose that the industrial plant decides to limit the probability of violating the 2-day regulation to 1%. What should be the daily mean CFR? Assume that the plant can reduce the mean daily CFR by improving its own in-house chemical process.
4.) Suppose that there is a new proposal in 2022 to change the regulation such that CFR will be measured for 4 days and that any manufacturing plant will be charged with a violation if the excessive CFR is observed in MORE THAN ONE of the 4 days. Which scheme is better (i.e., less strict) for the manufacturing plant? the 2-day scheme or the 4-day scheme? Justify your answer.
Thank you.
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