How can you find standard deviation from a probability distribution?

$"Standard deviation" = sqrt(E(X^2) - (E(X))^2)$

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

Dove $E(X) = int_(-oo) ^(oo) x *f(x) dx$

The variance is given by $Var(x) = E(X^2) - ( E(X) )^2$

Dove $E(g(X) ) = int_(-oo) ^(oo) g(x) * f(x) dx$

Sappiamo che

$"Standard Deviation" = sqrt( "Variance " )$

$=> "Standard deviation" = sqrt(E(X^2) - (E(X))^2)$

O...

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