Variance of Logistic Distribution/Proof 3
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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:
\(\ds \var X\) | \(=\) | \(\ds \frac 1 s \int_{-\infty}^\infty \dfrac {x^2 \map \exp {-\dfrac {\paren {x - \mu} } s} } {\paren {1 + \map \exp {-\dfrac {\paren {x - \mu} } s} }^2} \rd x - \mu^2\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 s \int_{-\infty}^\mu \dfrac {x^2 \map \exp {-\dfrac {\paren {x - \mu} } s} } {\paren {1 + \map \exp {-\dfrac {\paren {x - \mu} } s} }^2} \rd x + \frac 1 s \int_\mu^\infty \dfrac {x^2 \map \exp {-\dfrac {\paren {x - \mu} } s} } {\paren {1 + \map \exp {-\dfrac {\paren {x - \mu} } s} }^2} \rd x\ - \mu^2\) | Sum of Integrals on Adjacent Intervals for Integrable Functions | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 s \int_{-\infty}^0 \dfrac {\paren {sz + \mu}^2 \map \exp {-z} } {\paren {1 + \map \exp {-z} }^2} \paren {s\rd z } + \frac 1 s \int_0^\infty \dfrac {\paren {sz + \mu}^2 \map \exp {-z} } {\paren {1 + \map \exp {-z} }^2} \paren {s\rd z } - \mu^2\) | $z = \dfrac {\paren {x - \mu} } s$; $s\rd z = \rd x$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \int_{-\infty}^0 \dfrac {\paren {s^2z^2 + 2sz\mu} \map \exp {-z} } {\paren {1 + \map \exp {-z} }^2} \rd z + \int_0^\infty \dfrac {\paren {s^2z^2 + 2sz\mu} \map \exp {-z} } {\paren {1 + \map \exp {-z} }^2} \rd z\) | cleaning up; s and $\mu^2$ cancel | |||||||||||
\(\ds \) | \(=\) | \(\ds \int_0^\infty \dfrac {\paren {s^2z^2 - 2sz\mu} \map \exp {z} } {\paren {1 + \map \exp {z} }^2} \rd z + \int_0^\infty \dfrac {\paren {s^2z^2 + 2sz\mu} \map \exp {-z} } {\paren {1 + \map \exp {-z} }^2} \rd z\) | rewriting the first integral | |||||||||||
\(\ds \) | \(=\) | \(\ds \int_0^\infty \dfrac {\paren {s^2z^2 - 2sz\mu} \map \exp {z} } {\map \exp {2 z} \paren {1 + \map \exp {-z} }^2} \rd z + \int_0^\infty \dfrac {\paren {s^2z^2 + 2sz\mu} \map \exp {-z} } {\paren {1 + \map \exp {-z} }^2} \rd z\) | extracting $\map \exp {2 z}$ from the denominator of the first integral | |||||||||||
\(\ds \) | \(=\) | \(\ds 2s^2 \int_0^\infty \dfrac {z^2 \map \exp {-z} } {\paren {1 + \map \exp {-z} }^2} \rd z\) | simplifying |
From Sum of Infinite Geometric Sequence, we have:
- $\ds \sum_{n \mathop = 0}^\infty \paren {-1}^n \map \exp {-z}^n = \dfrac 1 {1 + \map \exp {-z} }$
Taking the derivative of both sides, we have:
\(\ds \sum_{n \mathop = 0}^\infty \paren {-n} \paren {-1}^n \map \exp {-z}^n\) | \(=\) | \(\ds \dfrac { \map \exp {-z} } {\paren {1 + \map \exp {-z} }^2}\) | ||||||||||||
\(\ds \sum_{n \mathop = 1}^\infty n \paren {-1}^{n + 1} \map \exp {-z}^n\) | \(=\) | \(\ds \dfrac { \map \exp {-z} } {\paren {1 + \map \exp {-z} }^2}\) | simplifying |
Therefore:
\(\ds \var X\) | \(=\) | \(\ds 2 s^2 \paren {\int_0^\infty z^2 \sum_{n \mathop = 1}^\infty n \paren {-1}^{n + 1} \map \exp {-z}^n \rd z}\) | substitution from above | |||||||||||
\(\ds \) | \(=\) | \(\ds 2 s^2 \paren {\sum_{n \mathop = 1}^\infty n \paren {-1}^{n + 1} \int_0^\infty z^2 \map \exp {-n z} \rd z}\) | Fubini's Theorem | |||||||||||
\(\ds \) | \(=\) | \(\ds 2 s^2 \paren {\sum_{n \mathop = 1}^\infty n \paren {-1}^{n + 1} \bigintlimits {-\dfrac 1 n \map \exp {-nz} \paren {z^2 + \frac {2 z} n + \frac 2 {n^2} } } 0 \infty }\) | Primitive of x squared by Exponential of a x | |||||||||||
\(\ds \) | \(=\) | \(\ds 2 s^2 \paren {\sum_{n \mathop = 1}^\infty n \paren {-1}^{n + 1} \paren {0 + \dfrac 1 n \paren {0^2 + \frac 0 n + \frac 2 {n^2} } } }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 4 s^2 \paren {\sum_{n \mathop = 1}^\infty \paren {-1}^{n + 1} \frac 1 {n^2} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 4 s^2 \paren {\dfrac {\pi^2} {12} }\) | Sum of Reciprocals of Squares Alternating in Sign | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {s^2 \pi^2} 3\) |
$\blacksquare$