We study the effect of a $Lambda$ hyperon immersed in the doubly magic nuclei, $^{16}$O, $^{40}$Ca, $^{48}$Ca, and $^{208}$Pb, as well as the neutron magic nucleus $^{90}$Zr. For a $Lambda$ in the $1s$ and $1p$ states in $^{17}_{Lambda}$O, $^{41}_{Lambda}$Ca, $^{49}_{Lambda}$Ca, $^{91}_{Lambda}$Zr, and $^{209}_{Lambda}$Pb, we compare the single-particle energies and density distributions of the core nucleons with those of the nuclei without the $Lambda$, as well as the point proton and neutron radii. A remarkable finding is that the bound $Lambda$ induces a significant asymmetry in the proton-neutron density distributions in the core nucleus. This in turn gives rise to an appreciable, iso-vector mean field. As a consequence, the neutrons in the core are more attracted to the center of the nucleus, while the protons are pushed away, in comparison with those in the corresponding nucleus without the $Lambda$.
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