Uniform Distribution Proofs

Property A: The moment generating function for the uniform distribution is

moment-generating-function-uniform

Proof:

Proof line 1

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

Proof (mean)

Now

Proof (expectation x^2)Thus

Proof (variance 1)

Proof (variance 2)

Property 1 of Order statistics from finite population: The mean of the order statistics from a discrete distribution is

Proof line 1

Proof: The proof is by induction on k.

Property 2 of Order statistics from continuous population: The pdf of the kth order statistic is

Continuous uniform distribution pdf

Proof: We use the fact that the pdf is the derivative of the cdf.

By Property 1 of Order statistics from continuous population, the cdf of the kth order statistic is

Continuous uniform distribution cdf

Now

f(x)

We now claim that the two sums in the last expression cancel each other out, leaving only the first expression, which is the desired result. To see that the difference between the last two sums is zero, make a change of variables in the last sum by replacing i by j-1. In this way the last sum becomes

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
No longer available online

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