Property A: The moment generating function for the uniform distribution is
Proof:
Property B: The mean for a random variable x with uniform distribution is (β–α)/2 and the variance is (β–α)2/12.
Proof:
Now
Thus
Property 1 of Order statistics from finite population: The mean of the order statistics from a discrete distribution is
Proof: The proof is by induction on k.
References
Ma D. (2010) The distribution of the order statistics. A Blog on probability and statistics
https://probabilityandstats.wordpress.com/2010/02/20/the-distributions-of-the-order-statistics/
Border, K. C. (2016) Lecture 15: Order statistics; conditional expectation. Caltech
https://healy.econ.ohio-state.edu/kcb/Ma103/Notes/Lecture15.pdf
Rundel, C. (2012) Lecture 15: order statistics. Duke University
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