## Drawing probability distribution

Almost regardless of your opinion about the predictability or efficiency of the markets, you will probably agree that for most assets, guaranteed returns are uncertain or risky. If we ignore the mathematics underlying probability distributions, we can see that they are pictures that describe a particular view of uncertainty. The probability distribution is a statistical calculation that describes the probability that a given variable will be between or within a specific range on a graph.

Uncertainty refers to randomness. It is different from the lack of predictability or the inefficiency of the market. An emerging research point of view holds that financial markets are uncertain and predictable. Furthermore, markets can be efficient but also uncertain.

In finance, we use probability distributions to draw pictures that illustrate our view of the sensitivity of an asset’s return when we think that the asset’s return can be considered a random variable. In this article, we’ll go over some of the most popular probability distributions and show you how to calculate them.

The distributions can be classified as discrete or continuous, and according to whether it is a probability density function (PDF) or a cumulative distribution.

## Discrete vs. Continuous Distributions

Discrete refers to a random variable drawn from a finite set of possible outcomes. A six-sided die, for example, has six discrete outcomes. A continuous distribution refers to a random variable drawn from an infinite set. Examples of continuous random variables include speed, distance, and some asset returns. A discrete random variable is typically illustrated with dots or dashes, while a continuous variable is illustrated with a solid line. The following figure shows discrete and continuous distributions for a normal distribution with mean (expected value) of 50 and a standard deviation of 10:

The distribution is an attempt to plot the uncertainty. In this case, a result of 50 is most likely, but it will only happen about 4% of the time; a result of 40 is one standard deviation below the mean and will occur just under 2.5% of the time.

## Probability density vs. cumulative distribution

The other distinction is between the probability density function (PDF) and the cumulative distribution function. The PDF is the probability that our random variable will reach a specific value (or in the case of a continuous variable, fall between an interval). We show that by indicating the probability that a random variable *X* will be equal to a real value *X:*