Normal and Non-normal Distributions. Here's a sample of your typical normal distribution curve plotted in red, and it is essentially just a summary of the bar charts in a histogram. A couple of key points, normal distribution is the most commonly found distribution, I'm sure we've all come across them, and they're often referred to as a bell curve, we've all heard that phrase for sure. And there are a couple of key attributes that must be present for a distribution to be considered normal. First of all, the mean, median, and mode of all the observations must be equal, and they must coincide with the highest point on the curve as seen. Also, the distribution is completely symmetrical, meaning that the both sides are identical, both to the left of the mean and to the right of the mean. And as a result, both sides of the mean contain 50% of the occurrences. Now, theoretically, while it's not so apparent on the chart, the distribution extends infinitely, both on the positive and on the negative sides, without touching zero. This is why you'll often hear a normal distribution called a continuous distribution. And lastly, the distribution can be completely described with just the average or the mean and the range of returns, a variance or standard deviation. Which means you're able to analyze large amounts of data relatively easy with just two measurements. And as a consequence, we can answer any probability question around a random variable if we know those two measurements, the mean and the variance or standard deviation. Let's go into an example. Before we dig into the details of example, let's point out a couple of things. First of all, every normal distribution has set probabilities between standard deviation intervals. For example, as you can see between -1 standard deviation and +1 standard deviation, a normal distribution will always contain 68.3% of the occurrences. So the probability within that range is 68.3%. Similarly, plus or minus two standard deviations is 95.5%, and plus or minus three standard deviations, we get almost all of the data or 99.7%. And because we know that the distribution is symmetrical and there's 50 off to the left of the mean, 50% to the left of the mean, and 50% to the right of the mean, we also know the intervals between one standard deviation, and two standard deviation, and three standard deviation on the positive and negative side. So from the mean to +1 standard deviation, as you can see, it contains 34.13%. Now, let's look at the details of our question. The average return of a stock is 15% with a standard deviation of 20%. Using the graphic below, what is the probability of the stock returning greater than 35%? Well, let's review what we know. First we know that the mean return is 15%. We also know that the standard deviation is 20%. So adding the mean to the standard deviation, 15% plus 20%, it equals 35%. And coincidentally, 35% is also the threshold that we're evaluating. We're looking for the probability of the stock returning greater than 35%. So essentially, we're calculating the area underneath the curve highlighted in blue. And again, given that each half equals 50%, we can add 50% to 34.13%, and subtract that from one to get the area under the curve shaded in blue, equaling 15.87%.

So the probability of the stock returning greater than 35% is 15.87%, assuming a normal distribution.

Now, not all distributions are normal, and we'll go through a couple of examples. Here we see your normal distribution with no skew, that means it's symmetrical, but a distribution can be skewed positively or it can be skewed on the negative side. Now, for a negative skew, despite having a higher probability of a smaller gain, probability of a significant loss is greater. Many short option strategies are negatively skewed given their uneven return stream. Here's another example. A distribution can be more peaked than a normal distribution, and that creates fatter tails than a normal distribution, and creates greater risk. You've probably heard the phrases fat tail problem or tail risks. Many of you all argue that hedge funds as an asset class have fatter tails and more tail risk than traditional investments, which means they have a greater probability of extreme values, both on the positive or negative side, but obviously we're more concerned on the negative side. And lastly, even if a return is not normal, modeling can still be done similar to what we did in the previous example. Modeling can still be done in most cases, but just with some added complexity.