Probability Distribution: Definition, Types, and Uses in Investing

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. Precisely where the possible value is likely to be plotted on the probability distribution depends on several factors, however. These factors include the distribution’s mean (average), standard deviation, skewness, and kurtosis.

How Probability Distributions Work

Perhaps the most common probability distribution is the normal distribution or bell curve although several distributions are commonly used. The data-generating process of some phenomenon will typically dictate its probability distribution. This process is referred to as the probability density function.

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

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

Discrete Probability Distribution Vs. Continuous Probability Distribution

Discrete and continuous probability distributions are two fundamental types of probability distributions, each describing different kinds of random variables. Understanding the differences between them is essential for correctly applying statistical methods and interpreting data.

Discrete probability distributions describe scenarios where the set of possible outcomes is countable and finite or countably infinite. These distributions are used when the random variable can take on specific, distinct values. For example, the number of heads in ten coin flips or the number of customers arriving at a store in an hour are cases of discrete random variables. In these scenarios, you can list all possible outcomes, such as 0, 1, 2, and so on. It's more likely that discrete probability distributions are more "choppy" since there are fewer outcomes.

In contrast, continuous probability distributions apply to random variables that can take on any value within a given range. These values are not countable because there are infinitely many possibilities within any interval. For example, the exact height of individuals in a population or the exact time it takes to complete a task are continuous variables. It's more likely that continuous probability distributions can have smoother distribution curves since there may be more outcomes.

Types of Probability Distributions

Probability distributions have many classifications. They include the normal distribution, chi-square distribution, binomial distribution, and Poisson distribution. These probability distributions serve different purposes and represent varying data generation processes.

Binomial

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

Another example would be to use a coin and figure out the probability of that coin coming up heads in 10 straight flips. A binomial distribution is discrete rather than continuous because only one or zero is a valid response.

Normal

The most commonly used distribution is the normal distribution. This is used frequently in finance, investing, science, and engineering. The normal distribution is fully characterized by its mean and standard deviation. The distribution isn't skewed and it does exhibit kurtosis.

This makes the distribution symmetric. It's 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.

Approximately 68% of the data collected in a normal distribution will fall within +/- one standard deviation of the mean. Approximately 95% will fall within +/- two standard deviations and 99.7% will fall within +/- three standard deviations. Unlike the binomial distribution, the normal distribution is continuous. All possible values are represented rather than just zero and one with nothing in between.

Poisson Distribution

The Poisson distribution is a discrete probability distribution that models the number of events occurring within a fixed interval of time or space. These events must happen independently of each other, and the average rate (mean number of occurrences) must be constant.

The key characteristic of the Poisson distribution is that it describes the probability of a given number of events happening within a specified interval when the events are rare and independent.

The Poisson distribution is used in various real-world applications where events occur randomly and independently. For example, it can model the number of customer arrivals at a bank in an hour, the number of emails received in a day, or the number of phone calls at a call center per minute.

Probability Distributions Used in Investing

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

The distribution of stock returns has been described as log-normal because stock prices are bounded by zero but offer a potentially unlimited upside. 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 the probability and time frame for a portfolio. An investor can also 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.

Probability Distribution and the Central Limit Theorem

The central limit theorem (CLT) is a statistical principle that states that the distribution of the sum of a large number of independent, identically distributed random variables approaches a normal distribution. This theorem matters because it allows statisticians to make inferences about population parameters even when the population distribution is unknown, as long as the sample size is sufficiently large.

One of the key implications of the CLT is that for large sample sizes, the sampling distribution of the sample mean will be approximately normally distributed. For instance, imagine you have a class of students where each student's height varies, but on average, they tend to be around 5 feet tall with some variability. According to the CLT, the distribution of average height samples will tend to follow a normal (bell-shaped) curve.

Example of a Probability Distribution

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 this image. Seven is the most common outcome (1+6, 6+1, 5+2, 2+5, 3+4, 4+3). Two and twelve are far less likely (1+1 and 6+6).

The Bottom Line

Probability distributions describe all of the possible values that a random variable can take. This 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.