We can use an amazing fact about the

normal distribution to define

*multidimensional normal distributions* in an elegant way. To pick a

*d*-

dimensional

normal distribution, just pick each

orthogonal coordinate independently according to a normal distribution (the coordinates may have different

standard deviations, if you like; the claims made in the

sequel still hold).

Obviously, this defines a distribution on **R**^{d}. What is less obvious is that this isn't dependent on the specific set of coordinates chosen! But it turns out that if you pick some other orthogonal basis, and take coordinates according to *that*, the projections of this distribution on each *new* coordinate are normally distributed, and *independent*!

This situation is truly impressive (for instance, if we used uniform distributions instead of normal ones, we'd still get a uniform distribution on some box in the first step; but the axes of the box would be parallel to the axes used, and any rotated set of axes would not have uniform or independent projections!