Bash : Check if PID exists with further logic

Question: How do you check if a PID exists and use the result within an IF statement in bash?

Things I've tried

if [ "$(ps -p $pid)" -eq 0 ]; then 
 echo "Running"
else
 echo "Not Running"
fi

---

if [ "$(kill -0 $pid)" -eq 0 ]; then 
 echo "Running"
else
 echo "Not Running"
fi

Neither of these evaluate correctly no matter how I redirect STDOUT/STDER