An ambulance travels back and forth, at a constant specific speed v, along a road of length L. We may model the location of the ambulance at any moment in time to be uniformly distributed over the interval (0, L). Also at any moment in time, an accident (not involving the ambulance itself) occurs at a point uniformly distributed on the road; that is, the accidents distance from one of the fixed ends of the road is also uniformly distributed over the interval (0, L). Assume the location of the accident and the location of the ambulance are independent. Supposing the ambulance is capable of immediate U-turns, compute the CDF and PDF of the ambulances travel time T to the location of the accident.
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!
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