Concept:
Group 15 hydrides contain a lone pair of electrons and hence are electron-rich. Heavy water (\(D_2O\)) has a greater density than ordinary water (\(H_2O\)). Water containing dissolved calcium and magnesium salts is called hard water.
Step 1: Examine Statement I.
Hydrides of Group 15 elements are:
\[
NH_3,\; PH_3,\; AsH_3,\; SbH_3
\]
All these molecules possess a lone pair of electrons on the central atom.
Hence they are electron-rich hydrides.
Therefore,
\[
\boxed{\text{Statement I is correct}}
\]
Step 2: Examine Statement II.
Heavy water contains deuterium instead of hydrogen.
\[
D_2O
\]
Density of heavy water:
\[
1.105\,g\,cm^{-3}
\]
Density of ordinary water:
\[
1.0\,g\,cm^{-3}
\]
Therefore,
\[
\boxed{\text{Statement II is correct}}
\]
Step 3: Examine Statement III.
Water containing dissolved salts of calcium and magnesium is known as hard water.
Such water does not form lather easily with soap.
Hence,
\[
\boxed{\text{Statement III is incorrect}}
\]
Thus, only statements I and II are correct.
\[
\boxed{\text{I, II only}}
\]