Suppose the distribution of speeds of cars traveling on the I-5 in California is nearly normal with a mean of 72.6 miles/hour and a standard deviation of 4.78 miles/hour. The speed limit on a stretch of the I-5 is 70 miles/hour. (a) What percentage of cars travel above the speed limit? (b) A highway patrol officer is hidden one the side of the freeway. What is the probability that 5 cars pass and none are speeding? Assume that the speeds of cars are independent of each other. (c) On average, how many cars would the highway patrol officer expect to watch until the first car that is speeding?
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!
Suppose the distribution of speeds of cars traveling on the I-5 in California is nearly normal with a mean of 72.6 miles/hour and a standard deviation of 4.78 miles/hour. The speed limit on a stretch of the I-5 is 70 miles/hour.
(a) What percentage of cars travel above the speed limit?
(b) A highway patrol officer is hidden one the side of the freeway. What is the probability that 5 cars pass and none are speeding? Assume that the speeds of cars are independent of each other.
(c) On average, how many cars would the highway patrol officer expect to watch until the first car that is speeding?
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