Note that these results need not hold if X and Y are not independent, are correlated. As an instructive example, note that it is possible to design an experiment where X and Y are distributed with significant variances as in Figures 9a-b,
but where there is no variance in their sum whatsoever, (
)!
An example of an apparatus generating these results would be a random number pair generator than just generates an internal random number Z with a significant variance but then reports to its two outputs X=-Z and Y=Z. (See Figure 10.)
While X and Y will have the same spread in their distributions as
Z, the sum X+Y will always be exactly zero and exhibit no spread
at all.
This is related to the observation that while there is a spread
when measuring the momentum p or the position r of the electron in the
ground state of a hydrogen atom, measuring the energy
, always gives precisely the same
result of 
.