Discrete Probability Distribution: Overview and Examples

What Is Discrete Distribution?

A discrete distribution is a probability distribution that depicts the occurrence of discrete (individually countable) outcomes, such as 1, 2, 3... or zero vs. one. The binomial distribution, for example, is a discrete distribution that evaluates the probability of a "yes" or "no" outcome occurring over a given number of trials, given the event's probability in each trial—such as flipping a coin one hundred times and having the outcome be "heads".

Statistical distributions can be either discrete or continuous. A continuous distribution is built from outcomes that fall on a continuum, such as all numbers greater than 0 (which would include numbers whose decimals continue indefinitely, such as pi = 3.14159265...). Overall, the concepts of discrete and continuous probability distributions and the random variables they describe are the underpinnings of probability theory and statistical analysis.

Key Takeaways

  • A discrete probability distribution counts occurrences that have countable or finite outcomes.
  • This is in contrast to a continuous distribution, where outcomes can fall anywhere on a continuum.
  • Common examples of discrete distribution include the binomial, Poisson, and Bernoulli distributions.
  • These distributions often involve statistical analyses of "counts" or "how many times" an event occurs.
  • In finance, discrete distributions are used in options pricing and forecasting market shocks or recessions.

Understanding Discrete Distribution

Distribution is a statistical concept used in data research. Those seeking to identify the outcomes and probabilities of a particular study will chart measurable data points from a data set, resulting in a probability distribution diagram. There are many types of probability distribution diagram shapes that can result from a distribution study, such as the normal distribution ("bell curve").

Statisticians can identify the development of either a discrete or continuous distribution by the nature of the outcomes to be measured. Unlike the normal distribution, which is continuous and accounts for any possible outcome along the number line, a discrete distribution is constructed from data that can only follow a finite or discrete set of outcomes.

Discrete distributions thus represent data that has a countable number of outcomes, which means that the potential outcomes can be put into a list. The list may be finite or infinite. For example, when studying the probability distribution of a die with six numbered sides the list is {1, 2, 3, 4, 5, 6}. A binomial distribution has a finite set of just two possible outcomes: zero or one—for instance, lipping a coin gives you the list {Heads, Tails}. The Poisson distribution is a discrete distribution that counts the frequency of occurrences as integers, whose list {0, 1, 2, ...} can be infinite.

A histogram of a binomial distribution
A histogram of a binomial distribution. C.K. Taylor

Distributions must be either discrete or continuous.

Examples of Discrete Distribution

The most common discrete probability distributions include binomial, Poisson, Bernoulli, and multinomial.

The Poisson distribution is also commonly used to model financial count data where the tally is small and is often zero. For one example, in finance, it can be used to model the number of trades that a typical investor will make in a given day, which can be 0 (often), or 1, or 2, etc. As another example, this model can be used to predict the number of "shocks" to the market that will occur in a given time period, say over a decade.

Another example where such a discrete distribution can be valuable for businesses is inventory management. Studying the frequency of inventory sold in conjunction with a finite amount of inventory available can provide a business with a probability distribution that leads to guidance on the proper allocation of inventory to best utilize square footage.

The binomial distribution is used in options pricing models that rely on binomial trees. In a binomial tree model, the underlying asset can only be worth exactly one of two possible values—with the model, there are just two possible outcomes with each iteration—a move up or a move down with defined probabilities.

Binomial Tree Model for Options Pricing
Binomial Tree Model for Options Pricing.

Image by Sabrina Jiang © Investopedia 2020

Discrete distributions can also be seen in the Monte Carlo simulation. Monte Carlo simulation is a modeling technique that identifies the probabilities of different outcomes through programmed technology. It is primarily used to help forecast scenarios and identify risks. In Monte Carlo simulation, outcomes with discrete values will produce discrete distributions for analysis. These distributions are used in determining risk and trade-offs among different items being considered.

Discrete Distribution FAQs

What Are the Types of Discrete Distribution?

The most common discrete distributions used by statisticians or analysts include the binomial, Poisson, Bernoulli, and multinomial distributions. Others include the negative binomial, geometric, and hypergeometric distributions.

What Are the Two Requirements for a Discrete Probability Distribution?

The probabilities of random variables must have discrete (as opposed to continuous) values as outcomes. For a cumulative distribution, the probability of each discrete observation must be between 0 and​ 1; and the sum of the probabilities must equal one (100%).

How Do You Know If a Distribution Is Discrete?

If there are only a set array of possible outcomes (e.g. only zero or one, or only integers), then the data are discrete.

What Is a Continuous Distribution?

Unlike a discrete distribution, a continuous probability distribution can contain outcomes that have any value, including indeterminant fractions. A normal distribution, for instance, is depicted by a bell-shaped curve with an uninterrupted line covering all values across its probability function.

What Is a Discrete Probability Model?

A discrete probability model is a statistical tool that takes data following a discrete distribution and tries to predict or model some outcome, such as an options contract price, or how likely a market shock will be in the next 5 years.

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