Assumptions and well-known facts:

1. are i.i.d. as

2. are constants.

Therefore and they are independent

Most importantly, jointly follows a multivariate normal distribution as they are independent.

As is an affine transformation of the multivariate normal vector, it has a normal distribution.

The question now comes:

What is ? Is it defined to be like which has non-zero mean?

The variables "in the second model" is another independent sample or what?

Are and independent?