Hi, I have two solutions to a problem one devised by me and the other by the instructor. They have the same result, but which is actually done correctly?

Problem:

Solution 1:We have an initial location with an uncertainty modelled with a Gaussian with . We take a GPS measurement which has an error variance .

Write the probabibility density functions of the prior and the measurement . And using Bayes rule, what is the posterior ?

Solution 2:

The specific calculations are not important, but I hope I've given enough information to say which is actually correct. Or maybe they both are, or neither?

(a normalizing constant)