I am trying to resolve the following expression:
\begin{equation} \int \dots \int \mathcal{N}(Z \mid w_0 + w_1 Y_1 + \dots + w_n Y_n, \sigma_w) \mathcal{N}(Y_1 \mid \mu_1, \sigma_1) \dots \mathcal{N}(Y_n \mid \mu_n, \sigma_n) dY_1 \dots dY_n \end{equation} where $Z, Y_1, \dots, Y_n$ are random variables distributed according to Gaussian distributions.
Intuitively I feel that the expression resolves to a Gaussian distribution, but I am not able to find the resolved distribution due to my limited knowledge in mathematics.
Can someone help me? Thanks in advance.