  The conventional probabilistic rounding error analysis in numerical linear
algebra provides worst-case bounds with an associated failure probability,
which can still be pessimistic. In this paper, we develop a new probabilistic
rounding error analysis from a statistical perspective. By assuming both the
data and the relative error are independent random variables, we derive the
approximate closed-form expressions for the expectation and variance of the
rounding errors in various key computational kernels. Our analytical
expressions have three notable characteristics: they are statistical and do not
involve a failure probability; they are sharper than other deterministic and
probabilistic bounds, using mean square error as the metric; they are correct
to all orders of unit roundoff and valid for any dimension. Furthermore,
numerical experiments validate the accuracy of our derivations and demonstrate
that our analytical expressions are generally at least two orders of magnitude
tighter than alternative worst-case bounds, exemplified through the inner
products. We also discuss a scenario involving inner products where the
underlying assumptions are invalid, i.e., input data are dependent, rendering
the analytical expressions inapplicable.
