What probability distribution calculators are available?

The collection includes nine distribution explorers: six discrete (binomial, geometric, negative binomial, Poisson, hypergeometric, uniform discrete) and three continuous (normal, exponential, uniform continuous). Each provides interactive parameter controls, probability calculations, and dynamic visualizations.

How do I choose which distribution to use?

Choose based on your data type and scenario. For counting successes in fixed trials, use binomial. For time between events, use exponential. For naturally occurring measurements that cluster around a mean, use normal. For rare event counts, use Poisson. Each explorer's page provides guidance on appropriate use cases.

What's the difference between discrete and continuous distributions?

Discrete distributions model countable outcomes (like number of heads in coin flips) and assign exact probabilities to specific values. Continuous distributions model measurable quantities (like height or time) and compute probabilities over intervals, not at single points.

What can I calculate with these distribution tools?

Each explorer calculates exact probabilities, cumulative probabilities (CDF), and summary statistics like mean and variance. You can adjust parameters interactively and see immediate visual feedback through PMF/PDF and CDF charts. All calculations update in real-time as you modify parameters.

Probability Distribution Explorers

Explore Probability Distributions

Interactive tools with dynamic visualizations, parameter controls, and probability calculators for discrete and continuous distributions.

Discrete Distributions

Binomial Distribution

Models the number of successes in a fixed number of independent trials

Geometric Distribution

Models the number of trials until the first success

Negative Binomial Distribution

Models the number of trials until a specified number of successes

Poisson Distribution

Models the number of events occurring in a fixed interval

Hypergeometric Distribution

Models sampling without replacement from a finite population

Uniform Discrete Distribution

Models equally likely outcomes over a discrete range

Continuous Distributions

Normal Distribution

The bell curve - models many natural phenomena

Exponential Distribution

Models time between events in a Poisson process

Uniform Continuous Distribution

Models equally likely outcomes over a continuous range

What are Probability Distributions?

Discrete vs Continuous Distributions

Discrete Distribution Explorers

Continuous Distribution Explorers

How to Use the Distribution Explorers

What are Probability Distributions?

A probability distribution describes how probabilities are distributed over the values of a random variable. It provides a mathematical function that gives the probabilities of occurrence of different possible outcomes. Every random variable has an associated probability distribution that characterizes its behavior.

For discrete random variables, the distribution assigns probabilities to specific, countable values. For continuous random variables, the distribution describes probability density over intervals. Understanding distributions is fundamental to statistics, data analysis, and probability theory.

Each distribution is characterized by parameters that determine its shape and behavior. For example, the binomial distribution uses

n

(number of trials) and

p

(probability of success), while the normal distribution uses μ (mean) and σ (standard deviation).

Discrete vs Continuous Distributions

Discrete distributions describe random variables that can only take on specific, countable values. Examples include the number of heads in coin flips, the number of defective items in a batch, or the number of customers arriving per hour. Key discrete distributions include discrete uniform, binomial, geometric, Poisson,negative binomial and hypergeometric.

Continuous distributions describe random variables that can take on any value within a range. Examples include height, temperature, time between events, or measurement errors. Key continuous distributions include normal, exponential, and uniform.

The mathematical treatment differs between types. Discrete distributions use probability mass functions (PMF) that assign exact probabilities to each value. Continuous distributions use probability density functions (PDF) where probabilities are calculated over intervals, not at single points.

Discrete Distribution Explorers

Our discrete distribution tools cover the most commonly used distributions in probability and statistics:

Binomial Distribution models the number of successes in a fixed number of independent trials, each with the same probability of success. Used in quality control, clinical trials, and survey sampling.

Geometric Distribution models the number of trials until the first success occurs. Commonly used in reliability testing and failure analysis.

Negative Binomial Distribution extends the geometric by modeling the number of trials needed to achieve a specified number of successes. Applications include epidemiology and insurance risk modeling.

Poisson Distribution models the number of events occurring in a fixed interval when events happen independently at a constant average rate. Used for arrival processes, defect counts, and rare event modeling.

Hypergeometric Distribution models sampling without replacement from a finite population. Essential for quality inspection and card game probabilities.

Uniform Discrete Distribution models equally likely outcomes over a discrete set, such as rolling a fair die.

Continuous Distribution Explorers

Our continuous distribution tools cover fundamental distributions for modeling real-valued random variables:

Normal Distribution is the most important distribution in statistics, describing data that clusters symmetrically around a mean. The bell curve appears in measurement errors, natural phenomena, and as a limiting distribution by the Central Limit Theorem. Used throughout statistics for hypothesis testing and confidence intervals.

Exponential Distribution models the time between events in a Poisson process. It describes waiting times, component lifetimes, and service times. The memoryless property makes it unique among continuous distributions.

Continuous Uniform Distribution models equally likely outcomes over a continuous interval. Every value in the range has the same probability density. Used as a baseline distribution and in random number generation.

Each continuous distribution explorer provides PDF visualization, CDF calculations, and probability computations for any interval.

How to Use the Distribution Explorers

Each interactive explorer provides a comprehensive toolkit for understanding and applying probability distributions:

Parameter Controls allow you to adjust distribution parameters using sliders or input fields. Watch the distribution shape change in real-time as you modify parameters.

Probability Calculations compute exact probabilities for specific values (discrete) or intervals (continuous). Calculate P(X = k), P(X ≤ k), P(X ≥ k), and custom ranges.

Visual Representations display the distribution through PMF/PDF charts and CDF curves. Color-coded regions highlight the probabilities you're calculating.

Summary Statistics show mean, variance, standard deviation, and other key properties. Understand how parameters affect these measures.

Experiment with different parameter values to build intuition about how each distribution behaves. Compare different distributions to understand when each is most appropriate for real-world applications.