# How do you calculate the standard deviation of a bounded probability distribution function?

Sep 2, 2016

$\sqrt{{\int}_{a}^{b} {x}^{2} \cdot f \left(x\right) \mathrm{dx} - {\left({\int}_{a}^{b} x f \left(x\right) \mathrm{dx}\right)}^{2}}$

#### Explanation:

Bounded or not use expectation, assume the following PDF
$f \left(x\right) , a < x < b$

then the expectation for standard deviation is calculated as

$\sqrt{V A R \left(x\right)} = \sqrt{E \left({x}^{2}\right) - E {\left(x\right)}^{2}} =$
$\sqrt{{\int}_{a}^{b} {x}^{2} \cdot f \left(x\right) \mathrm{dx} - {\left({\int}_{a}^{b} x f \left(x\right) \mathrm{dx}\right)}^{2}}$