# Uniform and Exponential Distributions

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A) A fire station is to be located along a road of length A, A<∞. If fires will occur at points uniformly chosen of (0, A), where should the station be located so as to minimize the expected distance from the fire? That is, choose a so as to

minimize E[|X-a|] when X is uniformly distributed over (0,A).

b) Now suppose that the road is of infinite length - stretching from point 0 outward to ∞. If the distance of a fire from point 0 is exponentially distributed with rate λ, where should the fire station now be located? That is, we want to

minimize E[|X-a|], where X is now exponential with rate λ.

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Uniform and exponential distributions are investigated. The solution is detailed and well presented. The response received a rating of "5" from the student who originally posted the question.

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