Let $x \in \mathbb{R}^n$ be random vector, $A \in \mathbb{R}^{m \times n}$ be a matrix and $b \in \mathbb{R}^m$ be a vector.
Now I should proof that the expected value is linear:
$$\mathbb{E}(Ax + b) = A \cdot \mathbb{E}(x) + b$$
The professor shared his proof. I understand every part of it, expcept for the first:
$$\mathbb{E}(Ax+b) = \int (Ax + b) f_x (x) \mathrm{d} x$$
Why is it not the following? Or is it the same?
$$\mathbb{E}(Ax+b) = \int (Ax + b) f_x (Ax + b) \mathrm{d} x$$
As explained by Did, I confused some variables. The correct first step would have been
$$\mathbb{E}(Ax+b) = \int u f_{Ax+b} (u) \mathrm{d} u$$
Due to the Law of the unconscious statistician (thank you Omnomnomnom) this is equal to
$$\int (Au+b) f_{x} (u) \mathrm{d} u$$