2002 Toyota Corolla diagnosis PO300

I am in the process of the diagnosis of a code PO300( random misfire)-I am using a scanner and the coolant temp sensor registers the same temp when cold as when warmed up fully- I just need a confirmation that the coolant sensor is not functioning causing a rough shaky idle when starting off in the am-after driving for 2-3 minutes in goes away and runs fine. My question is this the coolant sensor not functioning + possibly other issues?

Thanks in advance for info