# Variance of Logistic Distribution/Lemma 1

## Lemma for Variance of Logistic Distribution

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

## Proof

let:

 $\ds x$ $=$ $\ds \paren {1 - u}$ $\ds \leadsto \ \$ $\ds \frac {\d x} {\d u}$ $=$ $\ds -1$ Derivative of Power
 $\ds \int_{\to 0}^{\to 1} \map {\ln^2} {1 - u} \rd u$ $=$ $\ds -\int_{\to 1}^{\to 0} \map {\ln^2} x \rd x$ $\ds$ $=$ $\ds \int_{\to 0}^{\to 1} \map {\ln^2} x \rd x$ $\ds$ $=$ $\ds \bigintlimits {x \ln^2 x } 0 1 - 2 \int_{\to 0}^{\to 1} \map \ln x \rd x$ Primitive of Power of Logarithm of x $\ds$ $=$ $\ds \paren {\paren {0 - 0} - 2 \paren {-1} }$ Expectation of Logistic Distribution:Lemma 2 $\ds$ $=$ $\ds 2$

$\blacksquare$