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## Expected value of the smallest point in a plane

Hi,

Suppose i have n (two dimensional) points randomly chosen on a plane. Both x and y dimensions are intervals [0, 1).

What is the expected size of the kth smallest item?

Note: On a single dimension the answer is (to the best of my knowledge) k/(n+1).

thanks,

M.E.DALKILIC Ege University

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what do you mean by "size"? – suVRit Jun 25 at 9:18
The $k$th smallest is called the $k$th order statistic. en.wikipedia.org/wiki/Order_statistic Once you decide what you mean by size, you can use the probability distribution for the order statistics of the uniform distribution to write down the probability distribution for the order statistics of the sizes. The expected value is an integral related to this. I wouldn't expect a nice expression for this if the pdf and inverse of cdf for size are messy, such as for the Euclidean length, due to the nonsmooth point at length $1$. – Douglas Zare Jun 25 at 13:53