All available distributions
Continuous distributions
Discrete distributions
What is a probability distribution?
A probability distribution describes how probability is spread across the possible values of a random variable. It lets us quantify which outcomes are more likely, which are rare, and how a phenomenon behaves on average and in its variability.
Probability density function (PDF), probability mass function (PMF) and cumulative distribution function (CDF)
Continuous variables use the probability density function (PDF), which does not give a direct point probability but rather a density of probability around an interval. Discrete variables use the probability mass function (PMF), which does assign probability to each specific value.
The cumulative distribution function (CDF) applies to both cases and represents the cumulative probability up to a value x: \(P(X \le x)\). It is key for computing tails, intervals and percentiles.
Difference between continuous and discrete distributions
Discrete distributions: model integer counts (for example, the number of successes or events).
Continuous distributions: model magnitudes measured on a real scale (for example, times, lengths or measurement errors).
Brief explanation of each distribution
- Normal: continuous, symmetric and bell-shaped; the foundation of many models thanks to the central limit theorem.
- Binomial: discrete; the number of successes in n independent trials with constant probability p.
- Poisson: discrete; counts events in an interval when they occur at a constant average rate.
- Student's t: continuous; similar to the normal distribution but with heavier tails, useful for small samples with unknown variance.
- Chi-square: continuous and non-negative; widely used for inference about variances and goodness-of-fit/independence tests.
- Snedecor's F: continuous and non-negative; a ratio of variances, central to ANOVA and model comparison.
- Exponential: continuous; models the waiting time between events of a Poisson process.
- Uniform: continuous; every value within an interval has the same density.
- Gamma: continuous and positive; generalizes the exponential distribution and models cumulative waiting times.
- Beta: continuous on [0,1]; very useful for proportions and probabilities.
- Log-normal: continuous and positive; arises when the logarithm of the variable follows a normal distribution.
- Weibull: continuous and positive; common in reliability and survival/lifetime analysis.
- Negative binomial: discrete; the number of failures before observing a given number of successes.
- Geometric: discrete; the number of trials until the first success.
- Hypergeometric: discrete; sampling without replacement from a finite population.
- Bernoulli: discrete binary; a single trial with success (1) or failure (0).
How to choose a probability distribution
Start by identifying whether the variable is discrete or continuous. If you're counting successes in a fixed number of trials, you'll typically use the binomial distribution; if you're counting events in an interval with a stable rate, use the Poisson distribution; if you're working with continuous measurements around a mean, the normal distribution is usually the starting point.
Discrete data
Counts, success/failure outcomes, defects, arrivals or draws without replacement.
Continuous data
Measurements, waiting times, positive amounts, modeled proportions or transformed variables.
Critical values
For hypothesis tests and intervals, also check the statistics tables.
Worked example: choosing a distribution
A store receives an average of 12 orders per hour and wants to calculate the probability of receiving 18 or more in the next hour. The variable is a count of events in a fixed interval, so the right tool is the Poisson calculator with λ = 12.