# How can you find standard deviation from a probability distribution?

Feb 10, 2018

$\text{Standard deviation} = \sqrt{E \left({X}^{2}\right) - {\left(E \left(X\right)\right)}^{2}}$

#### Explanation:

In a PDF, $f \left(x\right)$ , the expected mean is given by $E \left(X\right)$

Where $E \left(X\right) = {\int}_{- \infty}^{\infty} x \cdot f \left(x\right) \mathrm{dx}$

The variance is given by $V a r \left(x\right) = E \left({X}^{2}\right) - {\left(E \left(X\right)\right)}^{2}$

Where $E \left(g \left(X\right)\right) = {\int}_{- \infty}^{\infty} g \left(x\right) \cdot f \left(x\right) \mathrm{dx}$

We know

 "Standard Deviation" = sqrt( "Variance " )

$\implies \text{Standard deviation} = \sqrt{E \left({X}^{2}\right) - {\left(E \left(X\right)\right)}^{2}}$

Or...

=> "Standard deviation" =sqrt( int_(-oo) ^(oo) x^2 *f(x) dx -( int_(-oo) ^(oo) x *f(x) dx)^2