Consider a random vector $X=[X_1, X_2, \ldots , X_n]$ where $X_i$ ($i \in 1, 2,\ldots, n$) are independent complex Gaussian random variables with zero mean and variance $\sigma_i^2$, i.e., $X_i \sim CN(0, \sigma_i^2)$.
How can I find expectation of norm of $X$, where the norm of $X$ is given by \begin{equation} \|X\|=\sqrt{\sum_{i=1}^n |X_i|^2} \end{equation}
Any help regarding this problem is really appreciated. Thanks.