  This paper establishes the optimal sub-Gaussian variance proxy for truncated
Gaussian and truncated exponential random variables. The proofs rely on first
characterizing the optimal variance proxy as the unique solution to a set of
two equations and then observing that for these two truncated distributions,
one may find explicit solutions to this set of equations. Moreover, we
establish the conditions under which the optimal variance proxy coincides with
the variance, thereby characterizing the strict sub-Gaussianity of the
truncated random variables. Specifically, we demonstrate that truncated
Gaussian variables exhibit strict sub-Gaussian behavior if and only if they are
symmetric, meaning their truncation is symmetric with respect to the mean.
Conversely, truncated exponential variables are shown to never exhibit strict
sub-Gaussian properties. These findings contribute to the understanding of
these prevalent probability distributions in statistics and machine learning,
providing a valuable foundation for improved and optimal modeling and
decision-making processes.
