Variance of Gaussian Distribution/Proof 1
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Theorem
Let $X \sim N \paren {\mu, \sigma^2}$ for some $\mu \in \R, \sigma \in \R_{> 0}$, where $N$ is the Gaussian distribution.
Then:
- $\var X = \sigma^2$
Proof
From the definition of the Gaussian distribution, $X$ has probability density function:
- $\map {f_X} x = \dfrac 1 {\sigma \sqrt {2 \pi} } \map \exp {-\dfrac {\paren {x - \mu}^2} {2 \sigma^2} }$
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:
\(\ds \var X\) | \(=\) | \(\ds \frac 1 {\sigma \sqrt {2 \pi} } \int_{-\infty}^\infty x^2 \map \exp {-\frac {\paren {x - \mu}^2} {2 \sigma^2} } \rd x - \mu^2\) | Expectation of Gaussian Distribution | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {\sqrt 2 \sigma} {\sigma \sqrt {2 \pi} } \int_{-\infty}^\infty \paren {\sqrt 2 \sigma t + \mu}^2 \map \exp {-t^2} \rd t - \mu^2\) | substituting $t = \dfrac {x - \mu} {\sqrt 2 \sigma}$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {\sqrt \pi} \paren {2 \sigma^2 \int_{-\infty}^\infty t^2 \map \exp {-t^2} \rd t + 2 \sqrt 2 \sigma \mu \int_{-\infty}^\infty t \map \exp {-t^2} \rd t + \mu^2 \int_{-\infty}^\infty \map \exp {-t^2} \rd t} - \mu^2\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {\sqrt \pi} \paren {2 \sigma^2 \int_{-\infty}^\infty t^2 \map \exp {-t^2} \rd t + 2 \sqrt 2 \sigma \mu \intlimits {-\frac 1 2 \map \exp {-t^2} } {-\infty} \infty + \mu^2 \sqrt \pi} - \mu^2\) | Fundamental Theorem of Calculus, Gaussian Integral | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {\sqrt \pi} \paren {2 \sigma^2 \int_{-\infty}^\infty t^2 \map \exp {-t^2} \rd t + 2\sqrt 2 \sigma \mu \cdot 0} + \mu^2 - \mu^2\) | Exponential Tends to Zero and Infinity | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {2 \sigma^2} {\sqrt \pi} \int_{-\infty}^\infty t^2 \map \exp {-t^2} \rd t\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac {2 \sigma^2} {\sqrt \pi} \paren {\intlimits {-\frac t 2 \map \exp {-t^2} } {-\infty} \infty + \frac 1 2 \int_{-\infty}^\infty \map \exp {-t^2} \rd t}\) | Integration by Parts | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {2 \sigma^2} {\sqrt \pi} \cdot \frac 1 2 \int_{-\infty}^\infty \map \exp {-t^2} \rd t\) | Exponential Tends to Zero and Infinity | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac{2 \sigma^2 \sqrt \pi} {2 \sqrt \pi}\) | Gaussian Integral | |||||||||||
\(\ds \) | \(=\) | \(\ds \sigma^2\) |
$\blacksquare$