Covariance of sum of multivariate normals If I have a variable $a = v*b + c$ where $v$ is a vector of length n, $b$ is a normally distributed random variable with mean 0 and variance 1, and $c$ is normally distributed with mean 0 and covariance matrix $\sigma^2*I$, is it possible to determine the covariance of $a$?
Note: I know that I can conclude that b and c are independent and uncorrelated
 A: The sum of two normal distributed variables is another normal distributed variable. This is also true for the multivariate case.
You can evaluate the solution most easily by regarding the characteristic functions.
$$\varphi_{X+Y}(t) = \varphi_X(t) \varphi_Y(t)$$
For the monovariate case this application of characteristic functions is explained here. For the multivariate case I can not find a link (and that is why I post this answer), but it will work the same.
If you add two multivariate normal distributions then the result will be a multivariate distribution with mean that is the sum of the two means and covariance matrix that is the sum of the two covariance matrices.

Heuristic derivation
The characteristic function is $$\varphi(\mathbf{t}) = \text{exp}\left(i\boldsymbol{\mu}^T\mathbf{t}-\frac{1}{2} - \mathbf{t}^T\boldsymbol{\Sigma}\mathbf{t}\right)$$
The product of two of them will become a sum inside the exponential term
$$\varphi(\mathbf{t}) = \text{exp}\left(i(\boldsymbol{\mu_1}+\boldsymbol{\mu_2})^T\mathbf{t}-\frac{1}{2} - \mathbf{t}^T(\boldsymbol{\Sigma_1}+\boldsymbol{\Sigma_2})\mathbf{t}\right)$$

Note that your $\mathbf{v}\cdot b$ is a multivariate normal distribution. It is a transformation of the multivariate normal distribution with a correlation matrix that has all ones $\boldsymbol{J}$ (all components are full correlated). Then you get a covariance matrix  of $\mathbf{V}\mathbf{J}\mathbf{V}^T$ where $\mathbf{V}$ is a diagonal matrix with the entries on the diagonal equal to $\mathbf{v}$.
(For the moment I do not know how to express this elegantly, but you can see this $\mathbf{v}\cdot b$ as a vector where the elements are fully correlated)
