If it rains real hard, is there a chance the rain's duration will be shorter? I think the if the rain in the clouds disperses with such force the rain will be over faster than if it was just a light drizzle. Can anyone confirm/deny this? Is there a percentage or study shown on rain duration relative to the rain's strength?
The intensity of a rainstorm does not actually cause the duration of the precipitation to be shorter. There is a strong correlation, but not in the sense you may be implying here (let me get back to that in moment).
First you have to dispel the model that a cloud is a big container of water, and if it "rains harder", the cloud will run out of water quicker and the rain event will be shorter in duration. Rain doesn't really work like that.
The atmosphere everywhere (even outside the cloud) contains lots and lots and lots of water. And under a stable combination of temperature, pressure, and humidity, all that moisture is perfectly happy to exist in a completely gaseous state. But as differing air masses (different temperatures, pressure, and humidity) start to collide, the local atmospheric conditions can fall outside the range where water can exist in a completely gaseous state. That is known as the dew point.
A cloud is where two differing atmospheric conditions have met, and the resulting condensed water droplets simply do not have sufficient weight to overcome the updraft speed of the air around it. If the air masses mix slowly, the rate of condensation will occur slowly and over a longer period of time. The resulting clouds may simply "evaporate" as the local conditions stabilize, or the condensing water may slowly precipitate over the ground (gentle rain), or the precipitation may never reach the ground at all (virga). These gentler atmospheric changes tend to play out over longer periods of time because the air masses are moving more slowly so it takes longer for the differences to mix and settle down into a new steady state.
Here's the part where I talk about intensity
If two air masses have widely varying conditions (think warm moist air over the ocean meeting dryer hotter conditions over the land) and they meet more abruptly (higher wind speeds), this causes a more-violent disturbance; the moisture from the air will condense more quickly, the rain falls faster and harder, and the whole thing ends (relatively speaking) more quickly.
But the duration tends to be shorter because the reaction between the differing masses is occurring at a faster rate — so the whole event will generally play out more quickly. But it is perfectly conceivable that "harder rain" can last much longer if the storm system continues to pull in more of the moist atmosphere around it. In extreme circumstances, that's pretty much what a hurricane is; the storm itself gets so big that it starts to create it's own system that pulls in more and more atmosphere around it. The rain doesn't simply stop when all the rain has fallen out of the original clouds. It actually (continually) makes new clouds/precipitation as it continues to draw in the atmosphere surrounding it.
But speaking in broad generalities, the faster the disturbance occurs, the faster the moisture will condense, the harder the precipitation will fall, the more quickly the conditions will stabilize, and the duration of the rain will be less.
Robert Cartanio's answer makes very good points, and I'll accentuate them with some examples.
Thunderstorms tend to produce "hard rain" and larger scale organized convection will have areas of hard rain and areas of weaker rain.
Air-mass thunderstorms are the type of daily convection you see in Florida and elsewhere. These tend to appear somewhat fast, rain out and be gone in short time. The other answer dispels the idea that the clouds rain themselves away. The reason for the short duration is that the rain cooled air underneath the storm blocks new moist warm air from entering the storm cloud, and it as the moisture is cut off, the storm dies.
A contrary example is a supercell thunderstorm, which will have extremely heavy precipitation and hail near the updraft. These storms can persist for hours, but your perception of how long the "heavy rain" lasts will be a function of the storm motion. A fast-moving supercell may pass by at 70 knots and you'll experience the heavy rain for a few minutes. A stationary supercell will give you the impression that the heavy rain lasts for an hour or longer. If you follow both of the storms they may have produced heavy rain for the same amount of time.
Similarly, a squall line will produce heavy precip with lighter precipitation behind it (usually, the stratiform rain can precede the convective rain or be laterally offset from it). How long this precip lasts (from your point of view) will depend on how fast it is moving. Regardless of how long you observe it for, it will likely last much longer as it moves. The stratiform region will be larger than the convective reason, so you will observe the lighter rain lasting longer than the heavy rain, but both will last for similar time scales.
In the cold season it is not uncommon to find stratiform rain, drizzle or snow that seems to last all day, and this might give the impression that light rain lasts a long time. In the hurricane season, strong convection can persist for weeks (though it might only observe a timescale of a day as it makes landfall where you live).
The point of all this is that the duration and intensity are strongly a function of the dynamics and your Eulerian point of view observing at a fixed location as a storm moves by. In the Lagrangian sense you'll find the correlation between duration and precipitation intensity is dependent on the dynamic and thermodynamic environment they are in.
It is likely true for the very heaviest rains, which exceed one inch per minute and last no more than a few minutes, that sustainable rainfall rate diminishes with increasing time that the rain lasts. New supplies of water vapor cannot be entrained into a cloud in such a short time, so such extreme rainfall rates can be sustained only until the water supply within the cloud is exhausted and the cloud rains itself out. But moderately heavy rains can be sustained for very long times. For example, rainfall rates averaging, say, 1 to 2 inches per hour (24 to 48 inches per day) can be sustained in the eye wall of a hurricane, perhaps even indefinitely, if the hurricane remains over the tropical ocean. The hurricane's inspiraling winds can continually replenish the water vapor supply fast enough to replace such a loss by rainfall in the eye wall. Of course, once a hurricane moves over colder, higher-latitude water or over land, the rate of replenishment is diminished and hence so is the sustainable rainfall rate.