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Home » From Boring to BOOM! The Hidden Power of JJJJ You Won’t Expect! - Abbey Badges

From Boring to BOOM! The Hidden Power of JJJJ You Won’t Expect! - Abbey Badges

📅 February 27, 2026 👤 scraface

Feb 27, 2026

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P(X_{(6)} - X_{(1)} \leq 2) \quad \text{for } X_i \sim \text{Uniform}[...
Scale: Let $ Y_i = X_i / 10 \sim \text{Uniform}[0,1] $. Then $ Y_{(1)}...
The joint distribution of $ (Y_{(1)}, Y_{(6)}) $ has joint density:
f_{Y_{(1)}, Y_{(6)}}(u,v) = 6 \cdot 5 \cdot (v - u)^{4}, \quad 0 \leq...
P(Y_{(6)} - Y_{(1)} \leq 0.2) = \int_{u=0}^1 \int_{v=u}^{u+0.2} 30(v -...
Change variable: let $ s = v - u $, so $ v = u + s $, $ 0 \leq s \leq...
But since $ u \leq 1 - s \leq 1 $, and $ s \leq 0.2 $, the upper limit...
\int_0^1 \int_0^{\min(0.2, 1 - u)} 30 s^4 \, ds \, du
Split into two regions: $ 0 \leq u \leq 0.8 $, where $ 1 - u \geq 0.2...
= \int_0^{0.8} \int_0^{0.2} 30 s^4 ds \, du + \int_{0.8}^1 \int_0^{1 -...

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