  We perform a detailed numerical study of the conductance $G$ through
one-dimensional (1D) tight-binding wires with on-site disorder. The random
configurations of the on-site energies $\epsilon$ of the tight-binding
Hamiltonian are characterized by long-tailed distributions: For large
$\epsilon$, $P(\epsilon)\sim 1/\epsilon^{1+\alpha}$ with $\alpha\in(0,2)$. Our
model serves as a generalization of 1D Lloyd's model, which corresponds to
$\alpha=1$. First, we verify that the ensemble average $\left\langle -\ln
G\right\rangle$ is proportional to the length of the wire $L$ for all values of
$\alpha$, providing the localization length $\xi$ from $\left\langle-\ln
G\right\rangle=2L/\xi$. Then, we show that the probability distribution
function $P(G)$ is fully determined by the exponent $\alpha$ and
$\left\langle-\ln G\right\rangle$. In contrast to 1D wires with standard
white-noise disorder, our wire model exhibits bimodal distributions of the
conductance with peaks at $G=0$ and $1$. In addition, we show that $P(\ln G)$
is proportional to $G^\beta$, for $G\to 0$, with $\beta\le\alpha/2$, in
agreement to previous studies.
