Continuous random linear transformation

If we have a continuous random variable, and we transform it by a linear function in the form mx+c, then we can apply some rules to find the continuous random expectation and continuous random variance of the transformed variable.

Expectation after transformation

If the expectation before the transformation was E(X), and we transform by the function mX+c, then:

E(mX+c)=m\times E(X)+c

You might be able to see then that, for any linear transformation of a continuous random variable, we just substitute the value of E(X) into the transformation function to get the expectation after the transformation:
t(E(X)) for a transformation function t(x).

See continuous random expectation transformation

Variance after transformation

If the variance before the transformation was Var(X), then, for a transformation of mX+c or mX, the transformed variance is the initial variance times the square of m.

Var(mX+c)=m^2\times Var(X)

See continuous random variance transformation