Variance of Logistic Distribution/Lemma 4
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Lemma for Variance of Logistic Distribution
- $\ds \int_{\to 0}^{\to 1} \map {\ln^2} {\dfrac {1 - u} u} \rd u = \dfrac {\pi^2} 3$
Proof
| \(\ds \int_{\to 0}^{\to 1} \map {\ln^2} {\dfrac {1 - u} u} \rd u\) | \(=\) | \(\ds \int_{\to 0}^{\to 1} \paren {\map {\ln} {1 - u} \rd u - \map {\ln} u}^2 \rd u\) | Difference of Logarithms | |||||||||||
| \(\ds \) | \(=\) | \(\ds \int_{\to 0}^{\to 1} \map {\ln^2} {1 - u} \rd u - 2 \int_{\to 0}^{\to 1} \map \ln {1 - u} \map \ln u \rd u + \int_{\to 0}^{\to 1} \map {\ln^2} u \rd u\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 2 - 2 \paren {2 - \dfrac {\pi^2} 6} + 2\) | Lemma 1, Lemma 2 and Lemma 3 | |||||||||||
| \(\ds \) | \(=\) | \(\ds \dfrac {\pi^2} 3\) |
$\blacksquare$