Conditional probability with normal distributions

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:

Quote:

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 1:

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?