Atsi arrives at a railway station at a random time. It is a hypothetical world and hence trains arrive exactly on time and the time between two successive trains is exactly 10 minutes. Atsi will wait and take the next train that arrives on the station after his arrival. Given that the time that Atsi arrives is uniformly random and the trains arrive 24 hours a day, 1. What is the mean time in minutes that Atsi must wait for the next train to arrive? What is the distribution of the waiting time of Atsi? 2. What is the probability that Atsi will have to wait at least 3 more minutes if he has already been waiting for 6 minutes? 3. In the real world, after the arrival of a train, the time until the next train arrives is an exponential random variable with a mean of 10 minutes. Atsi arrives at the station not knowing how long ago the previous train had come. What is the average time he must wait for the next train to arrive? What is the distribution of Atsi’s waiting time?
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!
Atsi arrives at a railway station at a random time. It is a hypothetical world and hence trains arrive exactly on time and the time between two successive trains is exactly 10 minutes. Atsi will wait and take the next train that arrives on the station after his arrival. Given that the time that Atsi arrives is uniformly random and the trains arrive 24 hours a day,
1. What is the
2. What is the probability that Atsi will have to wait at least 3 more minutes if he has already been waiting for 6 minutes?
3. In the real world, after the arrival of a train, the time until the next train
arrives is an exponential random variable with a mean of 10 minutes. Atsi
arrives at the station not knowing how long ago the previous train had
come. What is the average time he must wait for the next train to arrive?
What is the distribution of Atsi’s waiting time?
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