Expectation of Linear Transformation of Random Variable/Continuous

Theorem
Let $X$ be a continuous random variable.

Let $a, b$ be real numbers.

Then we have:


 * $\expect {a X + b} = a \expect X + b$

where $\expect X$ denotes the expectation of $X$.

Proof
Let $\map {\operatorname {supp} } X$ be the support of $X$.

Let $f_X : \map {\operatorname {supp} } X \to \R$ be the probability density function of $X$.

Then: