  In this work, we systematically investigate the favored $\alpha$-decay
half-lives and $\alpha$ preformation probabilities of both odd-$A$ and
doubly-odd nuclei related to ground and isomeric states around the doubly magic
cores at $Z=82$, $N=82$ and at $Z=82$, $N=126$, respectively, within a
two-potential approach from the view of the valence nucleon (or hole). The
results show that the $\alpha$ preformation probability is linear related to
$N_\text{p}N_\text{n}$ or $N_\text{p}N_\text{n}I$, where $N_\text{p}$,
$N_\text{n}$, and $I$ are the number of valence protons (or holes), the number
of valence neutrons (or holes), and the isospin of the parent nucleus,
respectively. Fitting the $\alpha$ preformation probabilities data extracted
from the differences between experimental data and calculated half-lives
without a shell correction, we give two analytic formulas of the $\alpha$
preformation probabilities and the values of corresponding parameters. Using
those formulas and the parameters, we calculate the $\alpha$-decay half-lives
for those nuclei. The calculated results can well reproduce the experimental
data.
