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discrete distribution

What is a discrete distribution?

A discrete distribution is a probability distribution that describes the occurrence of discrete (individually countable) outcomes, such as 1, 2, 3… or zero and one. For example, the binomial distribution is a discrete distribution that evaluates the probability of a “yes” or “no” outcome in a given number of trials, the probability of a given event in each trial – say a coin toss one hundred times, Then the result is “header”.

Statistical distributions can be discrete or continuous. Continuous distributions are constructed from results that fall on a continuum, such as all numbers greater than 0 (which includes numbers whose decimals are indefinitely continuous, such as pi = 3.14159265…). In general, the concepts of discrete and continuous probability distributions and the random variables they describe are fundamental to probability theory and statistical analysis.

key takeaways

  • Discrete probability distributions compute events with countable or finite outcomes.
  • This is in contrast to continuous distributions, whose results can fall anywhere on the continuum.
  • Common examples of discrete distributions include binomial, Poisson, and Bernoulli distributions.
  • These distributions usually involve statistical analysis of the “count” or “how many times” an event occurs.
  • In finance, discrete distributions are used to price options and predict market shocks or recessions.

Understanding Discrete Distributions

Distribution is a statistical concept used in data research. Those seeking to determine the outcomes and probabilities of a particular study will draw measurable data points from a dataset, resulting in a probability distribution map. Distribution studies can produce many types of probability distribution graph shapes, such as the normal distribution (“bell curve”).

Statisticians can identify the development of discrete or continuous distributions depending on the nature of the outcome to be measured. Unlike the normal distribution, which is continuous and can explain any possible outcome along the number line, the discrete distribution is constructed from data that can only follow a limited or discrete set of outcomes.

Therefore, a discrete distribution represents data with countable outcomes, which means that potential outcomes can be put into a list. The list may be limited or unlimited. For example, when studying the probability distribution of a dice with six numbered sides, the list is 1, 2, 3, 4, 5, 6. The binomial distribution has a finite set with only two possible outcomes: zero or one – for example, flipping a coin will give you the list Heads, Tails. The Poisson distribution is a discrete distribution that counts the frequency of occurrence as an integer whose list 0, 1, 2, … can be infinite.

A histogram of the binomial distribution.
CK Taylor

The distribution must be discrete or continuous.

Example 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 counts are small and often zero. For example, in finance, it can be used to simulate the number of trades a typical investor will make on a given date, which can be 0 (usually), 1, or 2, etc. As another example, this model can be used to predict the number of market “shocks” that will occur in a given time period, such as ten years.

Another example where this discrete distribution is valuable to a business is inventory management. Studying the frequency of selling inventory in conjunction with the limited amount of available inventory can provide businesses with a probability distribution that guides the correct allocation of inventory to best utilize square footage.

The binomial distribution is used in option pricing models that rely on binomial trees. In a binary tree model, the value of the underlying asset can only be one of two possible values—in this model, there are only two possible outcomes per iteration—moving up or down with a defined probability.

Binomial tree model for option pricing.

Discrete distributions can also be seen in Monte Carlo simulations. Monte Carlo simulation is a modeling technique that identifies the probabilities of different outcomes through programming techniques. It is primarily used to help predict scenarios and identify risks. In a Monte Carlo simulation, results with discrete values ​​produce discrete distributions for analysis. These distributions are used to determine the risks and trade-offs between the different projects under consideration.

Discrete Distribution FAQ

What are the types of discrete distributions?

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

What are the two requirements for discrete probability distributions?

The probability of a random variable must have discrete (rather than continuous) values ​​as a consequence. 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 the distribution is discrete?

If there is only one set of possible outcomes (such as only zero or one, or only integers), then the data is discrete.

What is a continuous distribution?

Unlike discrete distributions, continuous probability distributions can contain outcomes with any value, including indeterminate fractions. For example, the normal distribution is depicted by a bell-shaped curve with an unbroken line covering all the values ​​in its probability function.

What is a discrete probability model?

A discrete probability model is a statistical tool that takes discretely distributed data and attempts to predict or simulate certain outcomes, such as options contract prices, or the likelihood of market shocks over the next 5 years.

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