What Is a Discrete Distribution?

What Is a Discrete Distribution?

Let me explain what a discrete distribution is. It's a probability distribution that shows the likelihood of discrete, countable outcomes, like 1, 2, 3, or yes, no, true, false. Take the binomial distribution as an example—it's discrete and calculates the probability of yes or no results over multiple trials, based on the event's probability each time, such as getting heads in 100 coin flips.

You should know that statistical distributions come in discrete or continuous forms. Continuous ones deal with outcomes on a continuum, like any number greater than 0, including endless decimals like pi. Overall, discrete and continuous distributions, along with their random variables, form the foundation of probability theory and statistical analysis.

Key Takeaways

Discrete distributions differ from continuous ones because their outcomes are countable, not falling anywhere on a continuum. You'll see common examples like binomial, Poisson, and Bernoulli distributions. These often analyze counts or how many times events happen. In finance, they apply to options pricing and predicting market shocks or recessions.

Why Discrete Distributions Matter

A distribution is a statistical tool in data research. If you're trying to find outcomes and probabilities in a study, you chart data points to create a probability distribution diagram, which can take shapes like the normal bell curve.

You can tell if it's discrete or continuous by the outcomes' nature. Unlike the continuous normal distribution that covers any point on the number line, a discrete one uses data with only finite or discrete outcomes.

So, discrete distributions handle data with countable outcomes—you can list them and graph them. The list might be finite or infinite. For a six-sided die, the list is 1 through 6. With two dice, probabilities vary; rolling two sixes or two ones is rarer, shown by smaller bars on a graph.

Types of Discrete Probability Distributions

The main ones are binomial, Bernoulli, multinomial, and Poisson.

Binomial

This distribution has only two outcomes. Data from repeated trials get classified as success or failure, usually zero or one. Flipping a coin gives heads or tails.

It's used in options pricing with binomial trees, where an asset has just two possible values per step—up or down.

Bernoulli

Similar to binomial, but with one trial. Outcomes are zero (failure) or one (success). Think picking a green or red marble blindly and recording it that way. It's applied to assess if an investment succeeds or fails.

Multinomial

This covers more than two outcomes with counts. Like picking from green, red, yellow marbles and recording frequencies. In finance, these estimate probabilities of specific events.

Poisson Distribution

It shows the probability of a number of events in a fixed period. Counts are integers from 0 onward, potentially infinite. For example, picking marbles over two minutes and tracking successes.

It's used for financial data with small or zero tallies, like daily trades by an investor.

Monte Carlo Simulation

You also see discrete distributions in Monte Carlo simulations, which model probabilities of outcomes via programming to forecast scenarios and risks. Discrete values here produce discrete distributions for analyzing risks and trade-offs.

Calculation of Discrete Probability Distribution

Calculation varies by test and measurement. For two coin flips, possibilities are TT, HT, TH, HH—each 1/4, with HT/TH together at 1/2.

For two dice, there are 36 outcomes, from 2 to 12. Probabilities like P(X=2)=1/36, up to P(X=7)=6/36, and so on.

Investing Example

In a binomial tree model, starting at $10 with three-month intervals, probabilities are calculated for price rises or drops. Over nine months, it shows higher likelihood of price drops, like to $5.12 at 6.98 probability versus rises.

Discrete Distribution vs. Continuous Distribution

Discrete graphs use bars for countable variables. Continuous ones show curves for infinite values.

Frequently Asked Questions

What are the types? Binomial, Poisson, Bernoulli, multinomial, plus others like negative binomial, geometric, hypergeometric.

Requirements? Outcomes are discrete values; probabilities between 0 and 1, summing to 1.

How to know it's discrete? Only set possible outcomes, like integers.

What's continuous? Can have any value, like a bell curve.

What's a discrete probability model? Tool to predict outcomes like option prices or market shocks.

The Bottom Line

Discrete distributions graph finite outcomes like 1, 2, 3, true, false. Investors use them to estimate investing outcomes' likelihoods and choose hedging strategies accordingly.