## What Is a Probability Distribution?

A probability distribution is a statistical function that describes all the possible values and likelihoods that a random variable can take within a given range. This range will be bounded between the minimum and maximum possible values, but precisely where the possible value is likely to be plotted on the probability distribution depends on a number of factors. These factors include the distribution's mean (average), standard deviation, skewness, and kurtosis.

### Key Takeaways

- A probability distribution depicts the expected outcomes of possible values for a given data-generating process.
- Probability distributions come in many shapes with different characteristics, as defined by the mean, standard deviation, skewness, and kurtosis.
- Investors use probability distributions to anticipate returns on assets such as stocks over time and to hedge their risk.

## How Probability Distributions Work

Perhaps the most common probability distribution is the normal distribution, or "bell curve," although several distributions exist that are commonly used. Typically, the data-generating process of some phenomenon will dictate its probability distribution. This process is called the probability density function.

Probability distributions can also be used to create cumulative distribution functions (CDFs), which add up the probability of occurrences cumulatively and will always start at zero and end at 100%.

Academics, financial analysts, and fund managers alike may determine a particular stock's probability distribution to evaluate the possible expected returns that the stock may yield in the future.

The stock's history of returns, which can be measured from any time interval, will likely be composed of only a fraction of the stock's returns, which will subject the analysis to sampling error. By increasing the sample size, this error can be dramatically reduced.

## Types of Probability Distributions

There are many different classifications of probability distributions. Some of them include the normal distribution, chi-square distribution, binomial distribution, and Poisson distribution. The different probability distributions serve different purposes and represent different data generation processes.

### Binomial

The binomial distribution, for example, evaluates the probability of an event occurring several times over a given number of trials and given the event's probability in each trial. It may be generated, for example, by keeping track of how many free throws a basketball player makes in a game, where 1 = a basket and 0 = a miss.

Another typical example would be to use a fair coin and figure out the probability of that coin coming up heads in 10 straight flips. A binomial distribution is *discrete*, as opposed to continuous, since only 1 or 0 is a valid response.

### Normal

The most commonly used distribution is the normal distribution, which is used frequently in finance, investing, science, and engineering. The normal distribution is fully characterized by its mean and standard deviation, meaning the distribution is not skewed and does exhibit kurtosis.

This makes the distribution symmetric and it is depicted as a bell-shaped curve when plotted. A normal distribution is defined by a mean (average) of zero and a standard deviation of 1.0, with a skew of zero and kurtosis = 3.

In a normal distribution, approximately 68% of the data collected will fall within +/- one standard deviation of the mean; approximately 95% within +/- two standard deviations; and 99.7% within three standard deviations. Unlike the binomial distribution, the normal distribution is continuous, meaning that all possible values are represented (as opposed to just 0 and 1 with nothing in between).

## Probability Distributions Used in Investing

Stock returns are often assumed to be normally distributed but in reality, they exhibit kurtosis with large negative and positive returns seeming to occur more than would be predicted by a normal distribution.

In fact, because stock prices are bounded by zero but offer a potentially unlimited upside, the distribution of stock returns has been described as log-normal. This shows up on a plot of stock returns with the tails of the distribution having a greater thickness.

Probability distributions are often used in risk management as well to evaluate the probability and amount of losses that an investment portfolio would incur based on a distribution of historical returns.

One popular risk management metric used in investing is value-at-risk (VaR). VaR yields the minimum loss that can occur given a probability and time frame for a portfolio. Alternatively, an investor can get a probability of loss for an amount of loss and time frame using VaR. Misuse and overreliance on VaR have been implicated as one of the major causes of the 2008 financial crisis.

## Example of a Probability Distribution

As a simple example of a probability distribution, let us look at the number observed when rolling two standard six-sided dice. Each die has a 1/6 probability of rolling any single number, one through six, but the sum of two dice will form the probability distribution depicted in the image below. Seven is the most common outcome (1+6, 6+1, 5+2, 2+5, 3+4, 4+3). Two and twelve, on the other hand, are far less likely (1+1 and 6+6).

## What Makes a Probability Distribution Valid?

There are two steps to determining whether or not a probability distribution is valid. In step 1, the analysis should determine whether or not each probability is greater than or equal to zero and less than or equal to 1. In step 2, determine if the sum of all the probabilities is equal to 1. If both step 1 and step 2 are true, then the probability distribution is valid.

## How Are Probability Distributions Used in Finance?

There are two main ways in which probability distributions are used in Finance: (1) to estimate the returns of an investment asset and (2) to determine the possibility of loss events which will allow the investor to hedge their risk.

## What Are the Most Commonly Used Probability Distributions?

The most commonly used probability distributions are uniform, binomial, Bernoulli, normal, Poisson, and exponential.

## The Bottom Line

Probability distributions describe all of the possible values that a random variable can take. It is used in investing, particularly in determining the possible performance of a stock, as well as in the risk management component of investing by helping to determine the maximum loss.