1. What a probability distribution is and what it is for
Level 2: Distributions and why markets are not "normal"
The average height of a group of people is 1.70 m. That figure does not tell you whether almost everyone stands near it, or whether a lot of very short and very tall people share that same average.
Exactly the same happens with the returns of a strategy. And there the difference is paid in money.
The complete map, not two numbers
In the previous level you saw that a random variable is a rule assigning a number to each possible outcome of an uncertain experiment. You also saw that expectation and variance are two numbers summarizing its average behavior and its dispersion.
But one or two numbers never tell the whole story.
A probability distribution is the complete map. For every value the variable can take, it says how likely that value is to occur. If the variable is continuous, it says how densely outcomes cluster around each value.
It is the difference between knowing the average height of a group and having the full listing of how many people stand at each height. The second tells you whether most of them cluster near the average, or whether there are a lot of very short and very tall people sharing that same average.
Why it matters in trading
In trading, the variable you will care about most is the return of an asset over a given period: how much the price rises or falls, in percentage terms, from one day to the next or from one month to the next.
The distribution of those returns is, literally, the object you are trying to model every time you talk about risk, about the probability of a loss, or about expected return.
Here is the practical consequence. Two strategies can have the same average return and the same standard deviation and still behave very differently, because the shape of their return distribution is different.
One may earn a little on almost every day and lose a lot on a few days. The other may do exactly the opposite, while displaying the same two summary numbers.
Discrete and continuous
There are two broad families of distributions, depending on the type of variable.
Discrete distributions apply when the variable can only take a finite or countable set of values. The number of days the S&P 500 closes green during a 21-session month can only be 0, 1, 2 and so on up to 21, never 7.3.
Continuous distributions apply when the variable can take any value within a range. A stock's daily return, in principle, can be -3.47%, -3.471% or any figure in between. Financial returns are almost always treated as continuous variables.
For a discrete variable, the distribution can be written out as a list: each possible value alongside its probability, and those probabilities add up to 1, that is 100%, across all of them.
For a continuous variable the question changes. It makes no sense to ask what the probability is that the return equals exactly some number with infinitely many decimals, because that point probability is, in practice, zero: there are infinitely many possible values inside any range, however small.
What you do instead is talk about probability density. It is a curve such that the area beneath it between two points — say, between -1% and +1% — does give a real, meaningful probability: that the return falls in that range.
A distribution without formulas: the histogram
A very practical way to build a distribution without formulas is the histogram. The procedure is this:
- Take all the daily returns of a stock over, say, the last five years, roughly 1,250 market days.
- Group them into buckets: from -5% to -4%, from -4% to -3%, and so on.
- Count how many days fell into each bucket.
- Divide each count by the total number of days, and you get the relative frequency of each bucket.
That is an empirical approximation of the probability distribution of those returns. Empirical means based on real data, not on theory.
The more data you use and the finer the buckets, the more that histogram resembles the true underlying density curve.
Questions only the distribution can answer
The full distribution lets you answer questions that the mean and standard deviation alone cannot.
For example: if you buy this stock tomorrow, what is the probability of losing more than 3% in a single day?
With the mean and the variance you can make a rough estimate, but only if you assume the distribution has a specific shape, such as the normal you will see in the next module.
With the complete empirical distribution you can simply count. Of the last 1,250 days, how many saw a fall greater than 3%? That proportion is a direct estimate of the probability, with no need to assume any theoretical shape beforehand.
The cumulative: the 'up to here' version
It is also worth distinguishing the probability distribution from the cumulative distribution function.
The cumulative is simply the 'up to here' version. Instead of giving the probability of an exact value or of a narrow bucket, it gives the probability that the return is less than or equal to a given threshold.
A concrete example: if the cumulative function at -2% equals 0.10, that means 10% of historical days had a return of -2% or worse.
This cumulative function is the central piece for computing percentiles. And, as you will see in module 5, Value at Risk is defined exactly from it.
What comes next in this level
The rest of this level revolves, at bottom, around a single question: what shape does the distribution of financial returns actually have, and what happens if you use the wrong shape to make risk decisions?
You are going to study the normal distribution, the most widely used one thanks to its simplicity and mathematical tractability.
And then you are going to see why real market data departs from it systematically, precisely in the direction that hurts most: toward extreme losses that are more frequent than the tidy theory predicts.