Discrete Distribution
A discrete distribution is a probability distribution defined over a countable set of outcomes (for example: 0, 1, 2, …, or labels such as success/failure, yes/no). Each outcome has an associated probability; the probabilities are nonnegative and sum to 1. Discrete distributions are used to model counts and categorical events and contrast with continuous distributions, which assign probabilities over an uncountable continuum of values.
Key points
- Discrete outcomes are countable (finite or countably infinite); continuous outcomes form a continuum.
- Common discrete distributions: Bernoulli, binomial, multinomial, Poisson (plus negative binomial, geometric, hypergeometric).
- Discrete distributions are widely used in statistics, simulations (e.g., Monte Carlo), and finance (e.g., binomial option pricing, modeling rare events).
Common discrete distributions
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Bernoulli
A single trial with two outcomes: success (1) or failure (0). Used to model one-off yes/no events (e.g., whether an investment succeeds). -
Binomial
Number of successes in n independent Bernoulli trials with success probability p. PMF: P(X = k) = C(n, k) p^k (1 − p)^(n−k). Used in many areas including binomial tree models for option pricing. -
Multinomial
Generalization of the binomial to experiments with more than two categorical outcomes. Records counts for each category over n trials. -
Poisson
Models the count of events occurring in a fixed interval when events occur independently and at a constant average rate λ. PMF: P(X = k) = e^(−λ) λ^k / k!. Useful for modeling rare or low-count events (e.g., number of trades per day). -
Others
Negative binomial (counts until a fixed number of successes), geometric (trials until first success), hypergeometric (sampling without replacement).
How to calculate discrete probabilities
- Identify the sample space of countable outcomes and the event of interest.
- Assign probabilities to each outcome so that 0 ≤ P(X = x) ≤ 1 and Σ_x P(X = x) = 1.
- Use the appropriate probability mass function (PMF) for the chosen distribution (e.g., binomial or Poisson formulas above).
Examples:
– Coin flipped twice: sample space {HH, HT, TH, TT}. Each equally likely if fair coin: P(HT) = 1/4.
– Two dice summed: possible sums 2–12 with probabilities:
* P(2) = 1/36, P(3) = 2/36, P(4) = 3/36, P(5) = 4/36, P(6) = 5/36, P(7) = 6/36,
* P(8) = 5/36, P(9) = 4/36, P(10) = 3/36, P(11) = 2/36, P(12) = 1/36.
Discrete vs. continuous distributions
- Discrete: probabilities concentrated on separate points; visualized as bars (histogram or PMF).
- Continuous: probabilities spread over intervals; described by a probability density function (PDF) and shown as a curve. A continuous distribution assigns probability to ranges (areas under the curve), not to single points.
Applications and modeling
- Finance: binomial trees for option pricing, Poisson models for rare events (market shocks, low-frequency trades), and discrete-event simulations.
- Simulation: Monte Carlo simulations produce discrete distributions whenever inputs or outcomes take discrete values; those distributions help quantify risk and trade-offs.
How to recognize a discrete distribution
- Outcomes are a listable set (integers, categories).
- Probabilities are assigned to individual outcomes, not to intervals of real numbers.
- The sum of the probabilities over all possible outcomes equals 1.
Frequently asked questions
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What are the requirements for a discrete probability distribution?
Outcomes must be countable; each outcome’s probability must be between 0 and 1; the total probability across all outcomes must equal 1. -
When should I use a discrete model?
Use a discrete model when the data or event you’re modeling produces countable or categorical outcomes (e.g., number of defaults, counts of trades, categorical survey responses). -
What is a discrete probability model?
A statistical model that uses a discrete distribution (PMF) to predict or describe the probabilities of possible countable outcomes.
Bottom line
Discrete distributions model countable outcomes and are fundamental tools for analyzing categorical data, counts, and event frequencies. Choosing the correct discrete distribution and applying its PMF allows practitioners to estimate probabilities, conduct simulations, and make decisions under uncertainty.