is function convex?

Question

Suppose that $a_i$ has a normal distribution with mean $E(a_i)$ and variance $var(a_i)$ , we have for $\sum_{i=1}^n a_i x_i $ ,

$$E (\sum_{i=1}^n a_i x_i) = \sum_{i=1}^n E(a_i) x_i,$$

and

$$var (\sum_{i=1}^n a_i x_i)=\sum_{i=1}^n var(a_i) x_i ^2 +2\sum_{i=1}^{n-1} \sum_{j=i+1}^{n} cov(a_i,a_j)x_ix_j,$$

where $cov(a_i,a_j)x_ix_j$ is a covariance. Is this function convex ?

$$\sqrt{ \sum_i x_i ^2 var(a_i)+ 2 \sum_{i=1}^{n-1} \sum_{j=i+1}^{n}cov(a_i,a_j)x_ix_j + \sum_{i=1}^n E(a_i) x_i } $$

Answer 1

Note that if $a_1\sim \mathcal{N}(-1,1)$ then $$ \Delta = \sqrt{ var(a_1) x_1^2 + E(a_1) x_1}=\sqrt{x_1^2-x_1} $$ is not defined over a cone. Further, if $a_1\sim \mathcal{N}(0,1)$ $$ \Delta = \sqrt{ var(a_1) x_1^2 + E(a_1) x_1} = |x_1|. $$