# Probability Distribution: Definition, Types, and Uses in Investing

![Probability Distribution](https://www.investopedia.com/thmb/aSLWsokD5yKoR58mr--PevohRRY=/1500x0/filters:no_upscale():max_bytes(150000):strip_icc()/Terms-probability-distribution-4190874-35253427a7c4448a97bd8642b8684b21.jpg)

*— Madelyn Goodnight / Investopedia*

> **Definition**
>
> A probability distribution displays on a graph the statistical likelihood of every possible outcome that could occur within a specific time period.

## What Is a Probability Distribution?

A probability distribution, in statistics, determines the relative likelihood of each possible outcome that could occur within a set timeframe. Stock analysts use it to plot the likelihood of various movements in a stock's price, using data from its past performance.

Often displayed in a bell curve for ease of reference, probability distribution also could be used to project the minimum and maximum amount of resources that a business is likely to need on hand next year or to determine the likelihood of a severe outbreak of the flu versus a mild or moderate outbreak.

Probability distribution is a branch of mathematics that focuses on calculating the relative likelihood of every possible event that could occur within a set of variables.

### Key Takeaways

*   A probability distribution reveals the most probable and least probable occurrences based on a number of variables.
*   Investors use probability distributions to project the potential returns on assets such as stocks over time and to hedge their risk against potential losses.
*   Two of the fundamental types of probability distributions are discrete and continuous probability distributions.

## How Probability Distributions Work

Perhaps the most common probability distribution is the normal distribution or [bell curve](https://www.investopedia.com/terms/b/bell-curve.asp). 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 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.  

> **Important**
>
> A stock’s history of returns can be measured from any time interval but is usually limited to recent times. This can subject the analysis to sampling error, which can be reduced by increasing the sample size.

## 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 in which 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 that will turn up in 10 coin flips or the number of customers that could enter a store in an hour are cases of discrete random variables.

In these scenarios, you can list all possible outcomes, such as zero, one, two, and so on. Discrete probability distributions can be more "choppy" since there are fewer possible outcomes.

### Contrast Probability Distributions

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 infinite 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 will produce smoother distribution curves since more outcomes are possible.

## Types of Probability Distributions

Probability distributions have many classifications. They include the normal, [chi-square](https://www.investopedia.com/terms/c/chi-square-statistic.asp), [binomial](https://www.investopedia.com/terms/b/binomialdistribution.asp), 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.

For example, it could 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 figure out the probability of a coin coming up heads out of 10 straight flips. A binomial distribution is discrete rather than continuous because the only valid responses possible are one or zero.

### 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](https://www.investopedia.com/terms/k/kurtosis.asp). That is, the shape of the results when graphed indicates the degree of the differences observed, as in a bell curve.

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 one 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.

> **Fast Fact**
>
> Probability is the mathematical measure of the likelihood that an event will occur. It also refers to the branch of mathematics that concerns events and numerical descriptions of how likely they are to occur.

### 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 exhibit kurtosis, with large negative and positive movements occurring more often than would be predicted by a normal distribution.

The distribution of stock returns has been described as [lognormal](https://www.investopedia.com/articles/investing/102014/lognormal-and-normal-distribution.asp#:~:text=Because%20the%20values%20in%20a,create%20a%20right%2Dskewed%20curve.&text=This%20skewness%20is%20important%20in,lognormal%20distribution%20are%20normally%20distributed.) 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 to evaluate the probability and severity of losses that an investment portfolio could incur based on a distribution of historical returns.

One popular risk management metric used in investing is [value at risk](https://www.investopedia.com/terms/v/var.asp) (VaR). VaR yields the minimum loss that can occur given the probability and time frame for a portfolio.

Using VaR, an investor can get a probability of loss for an amount of loss and a time frame. It's worth noting that misuse and overreliance on VaR have been implicated as a major cause of the [2008 financial crisis](https://www.investopedia.com/articles/economics/09/financial-crisis-review.asp).

## 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 12 are far less likely (1+1 and 6+6).

![Image](https://www.investopedia.com/thmb/ZZVCAV-GFMuR7buksD9YfU3Baiw=/1500x0/filters:no_upscale():max_bytes(150000):strip_icc()/dotdash_Final_Probability_Distribution_Sep_2020-01-7aca39a5b71148608a0f45691b58184a.jpg)

*A role of the dice.*

## What Makes a Probability Distribution Valid?

A probability distribution is valid if two conditions are met:

*   Each probability is greater than or equal to zero and less than or equal to one.
*   The sum of all of the probabilities is equal to one.

## How Are Probability Distributions Used in Finance?

Probability distributions are [used in finance](https://www.investopedia.com/terms/a/additionruleforprobabilities.asp) for two purposes:

*   To estimate the likely returns on an investment
*   To estimate the possible extent of losses that must be hedged against if the investment is pursued.

## What Are the Most Commonly Used Probability Distributions?

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

## What Is the Difference Between Probability and Odds?

Probability measures the likelihood of an event occurring, expressed as a ratio of the number of favorable outcomes to the total number of possible outcomes.

Odds represent the ratio of the probability of an event happening to the probability of it not happening. For instance, if the probability of winning a game is 0.25, the odds are 1:3 (one win to three losses).

## What Is the Law of Large Numbers?

The law of large numbers states that as the number of trials or experiments increases, the average of the results obtained approaches the expected value or true probability. In short, its accuracy increases.

All studies rely on samples because of the impossibility of gathering data on an entire population. This principle assures that the sample mean converges to the population mean as more observations are collected, providing stability to statistical inference.

## The Bottom Line

Probability distributions describe all of the possible values that a random variable can take. This is used in investing, particularly for determining the possible performance of a stock and for evaluating the potential for losses.

> **Article Sources**
>
> 1.  PennState, Eberly College. "[STAT 500 Applied Statistics; Lesson 3: Probability Distributions](https://online.stat.psu.edu/stat500/book/export/html/470)."
>     
> 2.  Dartmouth Department of Mathematics. "[Grinstead and Snell’s Introduction to Probability](https://math.dartmouth.edu/~prob/prob/prob.pdf)." Pages 233-234.
>     
> 3.  PennState, Eberly College. "[STAT 500 Applied Statistics; 3.2.2 - Binomial Random Variables](https://online.stat.psu.edu/stat500/lesson/3/3.2/3.2.2)."
>     
> 4.  University of Pennsylvania. "[2.2.7 - The Empirical Rule.](https://online.stat.psu.edu/stat200/lesson/2/2.2/2.2.7)"
>     
> 5.  U.S. Securities and Exchange Commission. “[Remarks Before the Peterson Institute of International Economics](https://www.sec.gov/news/speech/remarks-peterson-institute-international-economics).”
>     
> 6.  Dartmouth Department of Mathematics. "[Grinstead and Snell’s Introduction to Probability](https://math.dartmouth.edu/~prob/prob/prob.pdf)."