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Home » \boxed\beginpmatrix 1 & 3 \\ 3 & 3 \endpmatrix - Dachbleche24

\boxed\beginpmatrix 1 & 3 \\ 3 & 3 \endpmatrix - Dachbleche24

📅 March 9, 2026 👤 scraface

Mar 09, 2026

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📌 P(X_{(6)} - X_{(1)} \leq 2) \quad \text{for } X_i \sim \text{Uniform}[0,10]

📌 Scale: Let $ Y_i = X_i / 10 \sim \text{Uniform}[0,1] $. Then $ Y_{(1)} \leq Y_{(6)} \leq 1 $, and condition becomes $ Y_{(6)} - Y_{(1)} \leq 0.2 $

📌 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 u \leq v \leq 1

📌 P(Y_{(6)} - Y_{(1)} \leq 0.2) = \int_{u=0}^1 \int_{v=u}^{u+0.2} 30(v - u)^4 \, dv \, du

📌 Change variable: let $ s = v - u $, so $ v = u + s $, $ 0 \leq s \leq 0.2 $, and $ s \leq 1 - u $

📌 But since $ u \leq 1 - s \leq 1 $, and $ s \leq 0.2 $, the upper limit $ 1 - u $ is automatically satisfied for $ u \geq 0.8 $, but we integrate $ u $ from 0 to 1.

📌 \int_0^1 \int_0^{\min(0.2, 1 - u)} 30 s^4 \, ds \, du

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