${\phi}_{i}$ is uniform random variable [${-\pi}, {\pi}$)
$$x =\sum _{ i }^{ }{ \frac { { a }_{ i } }{ { a }_{ 0 } } cos\left( { \phi }_{ i } \right) } $$
$$y =\sum _{ i }^{ }{ \frac { { a }_{ i } }{ { a }_{ 0 } } sin\left( { \phi }_{ i } \right) } $$
How can I calculate Expectation of the function below? $$E\left[ 3{ y }^{ 2 }{ x }^{ 2 }-\frac { 2 }{ 3 } { y }^{ 4 } \right] $$
Straight from definition, $$ \mathbb{E}[Z(x,y)] = \int_0^1 Z(x(\phi),y(\phi)) f_i(\phi) d\phi, $$ where $f(\cdot)$ is the pdf of $\phi_i$. Can you plug your functions in and evaluate?