Suppose we have a (possibly infinite) collection k-variate gaussian distributions $\{(\mathcal{N}(\mu_{\lambda}, \Sigma_{\lambda}))\}$ ($\lambda$ is just a label), and for each distribution $\mu \in [-1,1]^k$, the variance of each coordinate $X_i$ is $1$, and all other covariances $Cov[X_i ,X_j] \in [-1,1]$.

Suppose also we have a distribution $\Lambda$ over labels $\lambda$. Sample a vector $X \in \mathbb{R}^d$ by first sampling $\lambda \sim \Lambda$, and then sampling $X \sim \mathcal{N}(\mu_{\lambda}, \Sigma_{\lambda})$.

Now, sample $Y$ from $\mathcal{N}(\mathbb{E}_{\lambda \sim \Lambda}[(\mu_\lambda, \Sigma_\lambda)])$.

Are $X$ and $Y$ ever identically distributed?