What could be the relation between a bicycle's inner tube getting punctured and the pressure of the tube.
Are they directly proportional? I.e, if the pressure of Bicycle tube is higher then it has higher chance of getting punctured.
It's not going to be directly proportional. Leaving aside the quality of the tyre and what it will be rated for, the shape of the graph is likely to be a U shape:
It's not going to be as simple as that, but might work as a general rule.
The pressure has little effect on flats caused by sharp objects. Tire construction prevails here.
Pressure does have effect a big effect on pinch flats. The more pressure in your tire, the less it will deform when you hit a bump (or the very evil train tracks). If if can't deform enough to pinch the tube, no flat. As the pressure decreases, the amount of force required to pinch also decreases. I'll drop some pressure for rainy wet roads, but I'll be more careful about things that go bump when I do.
That said, don't exceed the manufacture's (both tire and rim) max or min pressure. Worth keeping in mind that 110 lbs put in during the cool morning will be quite a bit higher as the sun beats halfway through a four hour ride.
Conversely, your tube can lose pressure for many reasons as well. Fill it during a flat change with CO2 during a hot day and you may notice it being really low the next morning.
Tire pressure, should be checked prior to each ride.