# Variance of Logistic Distribution/Lemma 3

## Lemma for Variance of Logistic Distribution

$\ds \int_{\to 0}^{\to 1} \map \ln u \map \ln {1 - u} \rd u = 2 - \dfrac {\pi^2} 6$

## Proof

$\ds \ln \paren {1 - x} = -\sum_{n \mathop = 1}^\infty \dfrac {x^n} n$

Therefore:

 $\ds \int_{\to 0}^{\to 1} \map \ln u \map \ln {1 - u} \rd u$ $=$ $\ds \int_{\to 0}^{\to 1} \map \ln u \paren {-\sum_{n \mathop = 1}^\infty \dfrac {u^n} n } \rd u$ $\ds$ $=$ $\ds -\sum_{n \mathop = 1}^\infty \dfrac 1 n \int_{\to 0}^{\to 1} u^n \map \ln u \rd u$ Fubini's Theorem $\ds$ $=$ $\ds -\sum_{n \mathop = 1}^\infty \dfrac 1 n \bigintlimits {\dfrac {u^{n + 1} } {n + 1} \paren {\ln u - \dfrac 1 {n + 1} } } 0 1$ Primitive of Power of x by Logarithm of x $\ds$ $=$ $\ds -\sum_{n \mathop = 1}^\infty \dfrac 1 n \frac 1 {n + 1} \paren {0 - \frac 1 {n + 1} }$ $\ds$ $=$ $\ds \sum_{n \mathop = 1}^\infty \dfrac 1 n \dfrac 1 {\paren {n + 1}^2}$ $\ds$ $=$ $\ds \sum_{n \mathop = 1}^\infty \dfrac 1 n - \sum_{n \mathop = 1}^\infty \dfrac 1 {\paren {n + 1} } - \sum_{n \mathop = 1}^\infty \dfrac 1 {\paren {n + 1}^2}$ Partial Fractions Expansion: Reciprocal of x by x + 1 squared $\ds$ $=$ $\ds \sum_{n \mathop = 1}^\infty \dfrac 1 {n \paren {n + 1} } - \paren {\sum_{n \mathop = 1}^\infty \dfrac 1 {n^2} - 1}$ $\ds$ $=$ $\ds 1 - \paren {\dfrac {\pi^2} 6 - 1}$ Sum of Sequence of Reciprocals of Triangular Numbers and Basel Problem $\ds$ $=$ $\ds 2 - \dfrac {\pi^2} 6$

$\blacksquare$