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I have a problem similar to the one already described and solved here Mean minimum distance for N random points on a one-dimensional line:

In my case I have N random points on a line of length L populated with integers. I now select ONE random point of the N points. What is the expected distance to the nearest other point?

Cheers,

Peter

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I assume that your points are independent uniformly distributed. Then the probability that an interval of length $l$ around your point contains no other point is $(1-l/L)^{N-1}.$ I leave the rest as an exercise.

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