I am confused about the following expectations. Assume $z_n$ is an n-D random vector with multi-normal distribution and $a$ is an n-D real vector. Are these following two expectations identical? $$First: E_{z_n}(a^T z_n)^3 $$ $$Second: E_{a^T z_n}(a^T z_n)^3$$
And if they are different, how to compute $$E_{z_n}((a^T z_n)^2b^Tz_n)=?$$
Thank you very much.
PS: in my understanding, $$First: E_{z_n}(a^T z_n)^3=\int (a^T z_n)^3 f(z_n)dz_n $$ $$Second: E_{a^T z_n}(a^T z_n)^3=\int (a^T z_n)^3 g(a^T z_n)da^T z_n$$ where f(z_n) is the density function of $z_n$ and $g(a^T z_n)$ is that of $a^T z_n$.
I have been confused for years. Thank you for any kind help.