It's been years since I took probability, but why start from these particular set of assumptions?
You've shown that the correct form and constant factors falls out from assuming rotational invariance and independent components. But why is this particular set of assumptions intuitively what we call the "normal distribution"?
The same distribution should fall out starting from other more useful equivalent definitions right? (e.g., the maximum entropy distribution for some mean/variance, or the limiting distribution under CLT assumptions, etc)
You've shown that the correct form and constant factors falls out from assuming rotational invariance and independent components. But why is this particular set of assumptions intuitively what we call the "normal distribution"?
The same distribution should fall out starting from other more useful equivalent definitions right? (e.g., the maximum entropy distribution for some mean/variance, or the limiting distribution under CLT assumptions, etc)