In a PDF, f(x) f(x) , the expected mean is given by E(X) E(X)
Where E(X) = int_(-oo) ^(oo) x *f(x) dx E(X)=∫∞−∞x⋅f(x)dx
The variance is given by Var(x) = E(X^2) - ( E(X) )^2 Var(x)=E(X2)−(E(X))2
Where E(g(X) ) = int_(-oo) ^(oo) g(x) * f(x) dx E(g(X))=∫∞−∞g(x)⋅f(x)dx
We know
"Standard Deviation" = sqrt( "Variance " ) Standard Deviation=√Variance
=> "Standard deviation" = sqrt(E(X^2) - (E(X))^2) ⇒Standard deviation=√E(X2)−(E(X))2
Or...
=> "Standard deviation" =sqrt( int_(-oo) ^(oo) x^2 *f(x) dx -( int_(-oo) ^(oo) x *f(x) dx)^2 ⇒Standard deviation=√∫∞−∞x2⋅f(x)dx−(∫∞−∞x⋅f(x)dx)2
