Identify the correct statements.

The energy of a $K_{\beta}$ X-ray (transition from $M$ shell to $K$ shell) is greater than the energy of a $K_{\alpha}$ X-ray (transition from $L$ shell to $K$ shell). Since $E = \dfrac{hc}{\lambda}$, the wavelength of a $K_{\beta}$ X-ray is shorter than that of a $K_{\alpha}$ X-ray. Thus, statement (1) is incorrect.

A $K$ X-ray is emitted when an electron transitions from a higher energy shell (like $L$ or $M$) to a vacancy in the $K$ shell. This process can equivalently be described as a hole transitioning from the $K$ shell to a higher shell. Thus, statement (2) is correct.

The energy difference between the $K$ shell and higher shells is much larger than the energy difference between the $L$ shell and higher shells. Therefore, the energy of a $K$ X-ray is greater than the energy of an $L$ X-ray, which implies that the wavelength of a $K$ X-ray is shorter than that of an $L$ X-ray. Thus, statement (3) is correct.

An atom with a vacancy in an inner shell is in an excited state and therefore possesses more energy than a neutral atom in its ground state. Thus, statement (4) is incorrect.