Logically most normal people will think in terms of pull down , instead of pull-up, because we want everything to be OFF to be normal. and usually OFF means disconnected, which is why you would require a pull down, to ensure it is still off, but also connected.
The arduino is equipped with internal pull ups, not pull downs... Why is this? It is because It is more reliable to work with inverse logic for a lot of reasons, One important one in your case would be that the sensor needs to supply 5 volts to you to indicate everything is OK, and 0 volts when there is an alarm, because if the sensor fails (or a thief cuts the green wire, disconnecting your sensor
), it will indicate an alarm. if you use normal logic where OFF is no alarm, you wil never get an alarm if your sensor is dead.