Gaussian Distributions...

You are probably very familiar with the Gaussian distribution, means and standard deviations from the grading systems in your academic classes -- the infamous Bell Curve. When scoring in the low thirties on that third Orgo exam, the Gaussian distribution becomes our best friend because it demonstrates some type of average amid a wide range of samples.

Gaussians are one of many different types of probabilistic distributions. The subject of random processes and distributions is beyond the scope of our discussion (you'll encounter the topic again in quantum mechanics and physical chemistry) but suffice it to say that random processes can show trends in a characteristic fashion, known as a probability distribution. But what is a random process? A random process is a system described by a collection of random variables. And what are random variables? A random variable is a variable which can take on any of a known set of values. To bring this back to something a little more tangible, let's consider an example -- rolling two dice.

A die is a random variable. We have no way of knowing what value it will take, but we know that it can take only one of six values (1-6). It has an equal chance of taking each value. If we are rolling two dice, we know that there are only a certain number of sums that can be found (2-12), but there is no longer an equal chance of obtaining each outcome. However, since this process is still composed of random variables, we can mathematically predict the probability of different outcomes. This probability distribution is Gaussian in form:

The most probable, or average, outcome is the sum of 7. While this process may take on many different values, we know that 7 is most likely and this is graphically depicted as the peak of this distribution.