i cant quite tell what you are trying to write.
say if X is the random variable, then
then the expectation E[X] is the integration of x * pdf
and the variance is the integration of x^2 * pdf - E[X]^2
what if X^2 is the random variable?
what is the expectation and variance?
is Var [X^2] = integration of x^4 * pdf - E[X^2]^2?