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If the variables $\alpha_1$...$\alpha_n$ are distributed uniformly in $(0,1)$,

  1. How do I show that the spread $\alpha_{(n)}$ - $\alpha_{(1)}$ has density $n (n-1) x^{n-2} (1-x)$ and expectation $(n-1)/(n+1)$?
  2. What is the probability that all $n$ points lie within an interval of length $t$?
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3 Answers 3