Let $\mathbf{x}=[x_1, x_2,...,x_n]^T$ be a random vector, whose elements are independent random variables (parameters) $x_1, x_2, ..., x_n$. Probability density functions of $x_1, x_2, ..., x_n$ are given and they are not necessarily the same. Now let $f_1(\mathbf{x})\in\mathbb{R}, f_1(\mathbf{x})\in\mathbb{R},...,f_m(\mathbf{x})\in\mathbb{R}$ be functions of vector $\mathbf{x}$. Finally, let $g(\mathbf{x})$ be a product of functions $f_1(\mathbf{x}),f_2(\mathbf{x}),...,f_m(\mathbf{x})$, i.e. $g(\mathbf{x})=f_2(\mathbf{x})...f_m(\mathbf{x})$.
Is it possible to find the expected value of function $g(\mathbf{x})$ with respect to the vector $\mathbf{x}$? In other words, is it possible to calculate
$\mathbb{E}_{\mathbf{x}}\{g(\mathbf{x})\} = \mathbb{E}_{\mathbf{x}}\{f_2(\mathbf{x})...f_m(\mathbf{x})\}$ ?
Let me remind you that functions $f_1(\mathbf{x}),f_2(\mathbf{x}),...,f_m(\mathbf{x})$ are in $\mathbb{R}$, they are not vectors, while the expected value should be taken with respect to the vector $\mathbf{x}$. I have never seen something like this, so I wonder if this means that I need to successively find expected value of $g(\mathbf{x})$ with respect to each $x_i, i=1,...,n$, i.e. to calculate something like this
$\mathbb{E}_{\mathbf{x}}\{g(\mathbf{x})\} = \mathbb{E}_{x_n}\{\mathbb{E}_{x_{n-1}}\{\mathbb{E}_{x_{n-2}}\{...g(\mathbf{x})...\}\}\}$ ?