  Aims: This paper aims to derive an expression for the sensitivity of a
polarimetric radio interferometer that is valid for all-sky observations of
arbitrarily polarized sources, with neither a restriction on FoV nor with any a
priori assumption regarding the polarization state of the source. We verify the
resulting formula with an all-sky observation using the Murchison Widefield
Array (MWA) telescope. Methods: The sensitivity expression is developed from
first principles by applying the concept of System Equivalent Flux Density
(SEFD) to a polarimetric radio interferometer not by computing $A_e/T_{sys}$.
The SEFD is calculated from the standard deviation of the noisy flux density
estimate for a target source due to system noise. Results: The SEFD for a
polarimetric radio interferometer is generally not $1/\sqrt{2}$ of a
single-polarized interferometer as is often assumed for narrow FoV. This
assumption can lead to significant errors for a dual-polarized dipole based
system, which is common in low-frequency radio astronomy: up to $\sim 15\%$ for
a zenith angle (ZA) coverage of $45^\circ$, and up to $\sim45\%$ for $60^\circ$
coverage. The worst case errors occur in the diagonal planes of the dipole for
very wide FoV. This is demonstrated through theory, simulation and
observations. Furthermore, using the resulting formulation, calculation of the
off-zenith sensitivity is straightforward and unambiguous. Conclusions: For
wide FoV observations pertinent to low-frequency radio interferometer such as
the SKA-Low, the narrow FoV and the single-polarized sensitivity expressions
are not correct and should be replaced by the formula derived in this paper.
