Can You Buy The House of the Dead 4 for Xbox 360? No

No. The House of the Dead 4 (2005) was arcade-only (Sega Lindbergh hardware). No console port exists.

What Is on Xbox 360

• The House of the Dead: Overkill – Extended Cut (2011) – Rail shooter, grindhouse style, Move/light-gun support. Different game, same franchise.
• House of the Dead 2 & 3 Return (Wii / PS3) – Not on 360.

How to Play HOTD4 Today

• Sega Model 2/3 emulation – Not Lindbergh (different board).
• Lindbergh emulation – Experimental (Flycast / MAME WIP), high specs, no light-gun support.
• Original arcade cabinet – Collector route.

Stick with Overkill Extended Cut on 360 for a great light-gun experience.