  We consider an overdamped particle with a general physical mechanism that
creates noisy active movement (e.g., a run-and-tumble particle or active
Brownian particle etc.), that is confined by an external potential. Focusing on
the limit in which the correlation time $\tau$ of the active noise is small, we
find the nonequilibrium steady-state distribution
$P_{\text{st}}\left(\mathbf{X}\right)$ of the particle's position $\mathbf{X}$.
While typical fluctuations of $\mathbf{X}$ follow a Boltzmann distribution with
an effective temperature that is not difficult to find, the tails of
$P_{\text{st}}\left(\mathbf{X}\right)$ deviate from a Boltzmann behavior: In
the limit $\tau \to 0$, they scale as $P_{\text{st}}\left(\mathbf{X}\right)\sim
e^{-s\left(\mathbf{X}\right)/\tau}$. We calculate the large-deviation function
$s\left(\mathbf{X}\right)$ exactly for arbitrary trapping potential and active
noise in dimension $d=1$, by relating it to the rate function that describes
large deviations of the position of the same active particle in absence of an
external potential at long times. We then extend our results to $d>1$ assuming
rotational symmetry.
