Variance of Logistic Distribution/Proof 1
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Theorem
Let $X$ be a continuous random variable which satisfies the logistic distribution:
- $X \sim \map {\operatorname {Logistic} } {\mu, s}$
The variance of $X$ is given by:
- $\var X = \dfrac {s^2 \pi^2} 3$
Proof
From the definition of the logistic distribution, $X$ has probability density function:
- $\map {f_X} x = \dfrac {\map \exp {-\dfrac {\paren {x - \mu} } s} } {s \paren {1 + \map \exp {-\dfrac {\paren {x - \mu} } s} }^2}$
From Variance as Expectation of Square minus Square of Expectation:
- $\ds \var X = \int_{-\infty}^\infty x^2 \, \map {f_X} x \rd x - \paren {\expect X}^2$
So:
- $\ds \var X = \frac 1 s \int_{-\infty}^\infty \dfrac {x^2 \map \exp {-\dfrac {\paren {x - \mu} } s} } {\paren {1 + \map \exp {-\dfrac {\paren {x - \mu} } s} }^2} \rd x - \mu^2$
let:
\(\ds u\) | \(=\) | \(\ds \paren {1 + \map \exp {-\dfrac {\paren {x - \mu} } s} }^{-1}\) | Integration by Substitution | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \frac {\d u} {\d x}\) | \(=\) | \(\ds -\paren {1 + \map \exp {-\dfrac {\paren {x - \mu} } s} }^{-2} \paren {-\frac 1 s \map \exp {-\dfrac {\paren {x - \mu} } s} }\) | Power Rule for Derivatives, Chain Rule for Derivatives and Derivative of Exponential Function: Corollary 1 | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \dfrac 1 u - 1\) | \(=\) | \(\ds \paren {\map \exp {-\dfrac {\paren {x - \mu} } s} }\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \map \ln {\dfrac 1 u - 1}\) | \(=\) | \(\ds -\dfrac {\paren {x - \mu} } s\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds -s \map \ln {\dfrac {1 - u} u} + \mu\) | \(=\) | \(\ds x\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \mu^2 -2 s \mu \map \ln {\dfrac {1 - u} u} + s^2 \map {\ln^2} {\dfrac {1 - u} u}\) | \(=\) | \(\ds x^2\) |
and also:
\(\ds \lim_{x \mathop \to -\infty} \paren {1 + \map \exp {-\dfrac {\paren {x - \mu} } s} }^{-1}\) | \(=\) | \(\ds 0\) | ||||||||||||
\(\ds \lim_{x \mathop \to \infty} \paren {1 + \map \exp {-\dfrac {\paren {x - \mu} } s} }^{-1}\) | \(=\) | \(\ds 1\) |
Then:
\(\ds \var X\) | \(=\) | \(\ds \int_{\to 0}^{\to 1} \paren {\mu^2 - 2 s \mu \map \ln {\dfrac {1 - u} u} + s^2 \map {\ln^2} {\dfrac {1 - u} u} } \rd u - \mu^2\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \mu^2 \int_{\to 0}^{\to 1} \rd u - 2 s \mu \int_{\to 0}^{\to 1} \map \ln {\dfrac {1 - u} u} \rd u + s^2 \int_{\to 0}^{\to 1} \map {\ln^2} {\dfrac {1 - u} u} \rd u - \mu^2\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \mu^2 - 2 s \mu \paren {\int_{\to 0}^{\to 1} \map \ln {1 - u} \rd u - \int_{\to 0}^{\to 1} \map \ln u \rd u} + s^2 \paren {\int_{\to 0}^{\to 1} \map {\ln^2} {1 - u} \rd u - 2 \int_{\to 0}^{\to 1} \map \ln {1 - u} \map \ln u \rd u + \int_{\to 0}^{\to 1} \map {\ln^2} u \rd u} - \mu^2\) | Definite Integral of Constant and Difference of Logarithms | |||||||||||
\(\ds \) | \(=\) | \(\ds \mu^2 - 2 s \mu \paren {\paren {-1} - \paren {-1} } + s^2 \paren {2 - 2 \paren {2 - \dfrac {\pi^2} 6} + 2} - \mu^2\) | Expectation of Logistic Distribution:Lemma 1, Expectation of Logistic Distribution: Lemma 2, Lemma 1, Lemma 2 and Lemma 3 | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {s^2 \pi^2} 3\) |
$\blacksquare$