Decomposition of Multivariate Normal distribution with zero covariance

Question

Is the following true?

N([$s_1, s_2$] | [0,0], [[$\sigma$, 0],[0, $\sigma$]]]) = N($s_1$ | 0, $\sigma$) * N($s_2$ | 0, $\sigma$)

Answer 1

This is not true in the general case. Using the definition of the MVN distribution you can prove that,

N([s1,s2] | [0,0], [[σ, 0],[0, σ]]]) = σ N(s1$\sqrt(σ)$ | 0, σ) N(s2$\sqrt(σ)$ | 0, σ).