PIR sensor (with open collector) + Arduino - anomalous results
These devices require (typically) 10-24v to operate reliably due to the onboard regulator. With a supply as low as 5v all sorts of stability issues will occur.

Try powering the PIR with 12v connected as its supply. Retain the 10k resistor though as this is performing the open collector level conversion and giving an output that will swing from 0-5v. Interestingly, if this were connected to the Arduino 3.3v terminal then the ouput level will swing from 0v to 3.3v and this is the beauty of of using open collector devices. They can allow compatibility between circuits of different logic and supply levels.

Have a look at this useful resource for the explanation of "open collector" devices.

http://www.evilmadscientist.com/2012/basics-open-collector-outputs/