Middle order statistic given the min and max of four

Question

Four i.i.d. Uniform $(0,1)$ variables are sorted as $O_1\le O_2\le O_3\le O_4$ . Given $O_1=0.3$ and $O_4=0.9$ , find $\mathbb{E}[O_3]$ .

Short answer

$0.7$ . Conditionally the two middle values are i.i.d. uniform on $(0.3,0.9)$ ; their larger one sits on average two-thirds of the way along, at $0.3+\tfrac23(0.6)=0.7$ .

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Lessons behind this question

Order Statistics and Sample Quantiles
Expectation, Mean, and Variance
Conditional Expectation and Variance Revisited

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