Editing Variance (section)

===Absolutely continuous random variable===

If the random variable <math>X</math> has a [[probability density function]] <math>f(x)</math>, and <math>F(x)</math> is the corresponding [[cumulative distribution function]], then

<math display="block">\begin{align}
   \operatorname{Var}(X) = \sigma^2 &= \int_{\R} {\left(x - \mu\right)}^2 f(x) \, dx \\[4pt]
     &= \int_{\R} x^2 f(x)\,dx -2\mu\int_{\R} xf(x)\,dx + \mu^2\int_{\R} f(x)\,dx \\[4pt]
     &= \int_{\R} x^2 \,dF(x) - 2 \mu \int_{\R} x \,dF(x) + \mu^2 \int_{\R} \,dF(x) \\[4pt]
     &= \int_{\R} x^2 \,dF(x) - 2 \mu \cdot \mu + \mu^2 \cdot 1 \\[4pt]
     &= \int_{\R} x^2 \,dF(x) - \mu^2,
 \end{align}</math>

or equivalently,

<math display="block">\operatorname{Var}(X) = \int_{\R} x^2 f(x) \,dx - \mu^2 ,</math>

where <math>\mu</math> is the expected value of <math>X</math> given by

<math display="block">\mu = \int_{\R} x f(x) \, dx = \int_{\R} x \, dF(x). </math>

In these formulas, the integrals with respect to <math>dx</math> and <math>dF(x)</math>
are [[Lebesgue integral|Lebesgue]] and [[Lebesgue–Stieltjes integration|Lebesgue–Stieltjes]] integrals, respectively.

If the function <math>x^2f(x)</math> is [[Riemann-integrable]] on every finite interval <math>[a,b]\subset\R,</math> then

<math display="block">\operatorname{Var}(X) = \int^{+\infty}_{-\infty} x^2 f(x) \, dx - \mu^2, </math>

where the integral is an [[improper Riemann integral]].