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Jul 2, 2022 at 13:28 history edited Maverick Meerkat CC BY-SA 4.0
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Jun 30, 2022 at 8:36 comment added Steve Kroon I think it is a bit of both: LOTUS says that we can change the parameterization of the expectation. Your argument shows that when the parameterization is changed for the Gaussian (or any location-scale) family, the resulting density for the new parameterization will be in the same family, as noted in the question: "because the Gaussian distribution is a member of the location-scale family: standardizing a Gaussian gives another Gaussian distribution."
Jun 27, 2022 at 13:34 history answered Maverick Meerkat CC BY-SA 4.0