Can I detect with software when a PIR sensor is physically connected to Arduino?
The PIR sensor I have will overpower the internal pull-up resistor of the ATmega328P used on the Arduino Nano. This means if you turn on the pull-up and then get a read of LOW you know something is connected to that pin because the unconnected pin will always read HIGH with the pull-up on. With this approach you can get false negatives because the pin will also read HIGH if the PIR is triggered but you won't ever get false positives for PIR sensor detection. This means you can't rely on just doing a single test in setup() because the PIR might be triggered then so you will need to repeatedly run the test until the PIR sensor is detected. Example sketch:

int PIRpin = 2;    //the input pin for the PIR sensor
boolean PIRconnected = false;

void setup() {
  Serial.begin(9600);
  pinMode(PIRpin, INPUT);
}

void loop() {
  if (PIRconnected == false) {
    pinMode(PIRpin, INPUT_PULLUP);
    if (digitalRead(PIRpin) == LOW) {
      Serial.println(F("PIR sensor detected"));
      PIRconnected = true;
    }
    pinMode(PIRpin, INPUT);  //turn off the internal pull-up to reduce power consumption
  }
  Serial.println(digitalRead(PIRpin));
  delay(500);
}