Covariance of a Point between 2 Points with Known Covariances

Question

If two points $p_1$ and $p_2$ observed in a plane are each associated with a known covariance matrix, is it possible to infer the covariance matrix of any point on the segment $[p_1,p_2]$?

Answer 1

The question is if we can compute the covariance matrix of the random vector

$$\lambda p_1+(1-\lambda)p_2$$

assuming that we know the covariance matrices of $p_1$ and $p_2$.

By definition, the elements of the covariance matrix in question has to be calculated as

$$E((\lambda p_{1,i}+(1-\lambda )p_{2,i})(\lambda p_{1,j}+(1-\lambda )p_{2,j})]$$

where $p_{k,l}$ are the components of $p_1$ and $p_2$ ($k=1,2; l=1,2,...,n$).

In the product above the following terms appear:

$$E[p_{1,i}p_{2,j}]\,\,\text{ and }\,\,E[p_{2,i}p_{1,j}]$$ which are unknown to us if we know nothing about the common distribution of $p_1$ and $p_2$.