When crossing the Golden Gate Bridge traveling into San Francisco, all drivers must pay a toll. Suppose the amount of time (in minutes) drivers wait in line to pay the toll follows an exponential distribution with a probability density function of f(x) = 0.2e-0.2x a. What is the mean waiting time that drivers face when entering San Francisco via the Golden Gate Bridge? Mean waiting time b. What is the probability that a driver spends more than the average time to pay the toll? (Round intermediate calculations to at least 4 decimal places and final answer to 4 decimal places.) Probability
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 waiting time drivers in line to pay the toll follows exponential distribution with probability function as,
f(x) = 0.2e-0.2x
Part a:
We have, θ = 0.2
The mean waiting time that drivers face when entering the San Francisco via golden gate is 1/θ = 1/0.2 = 5 minutes.
Part b:
The probability that a driver spends more than the average time to pay the toll is computed as,
Thus, the probability that a driver spends more than the average time to pay the toll is 0.3679.
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