Probability Distributions in Trading: An Overview

Probability Distributions Explained

Probability distributions describe the range and likelihood of outcomes. Learn the main ones used in trading, their shapes and when each applies.

Advanced3 min readUpdated 3 Oct 2026

Lesson 22 of 46

A probability distribution describes all the possible values of an uncertain quantity and how likely each one is. Daily returns, the number of winning trades in a month, the time between large market moves and the outcome of a prediction market all follow some distribution. Choosing the right distribution matters: assuming returns are normal when they have fat tails can badly understate the risk of large losses. This lesson gives an overview; the following lessons cover the most important distributions in detail.

Discrete vs continuous distributions#

Type	Describes	Examples
Discrete	Countable outcomes	Number of wins in 20 trades, number of large moves in a month
Continuous	Any value in a range	Returns, prices, volatility, time until an event

Discrete distributions use a probability mass function (probability of each value); continuous ones use a probability density function (probability of ranges of values).

Distributions used in trading#

Distribution	Typical use	Lesson
Normal (Gaussian)	Simple model of returns; errors; averages	Normal Distribution
Lognormal	Prices that cannot go below zero; Black Scholes	Lognormal Distribution
Student's t	Fat tailed returns; small sample tests	Student's t-Distribution
Binomial	Number of wins in a fixed number of trades	Binomial and Bernoulli Distributions
Poisson	Number of rare events in a period	Poisson and Exponential Distributions
Bernoulli	Single yes or no outcome, such as a prediction market	Random Variables
Exponential	Time between events
Uniform	Equal chance across a range; random number generation
Mixtures	Returns from calm and volatile regimes	Empirical and Mixture Distributions
Power laws and extreme value distributions	The largest losses in the tails	Fat Tails

Describing a distribution#

Measure	What it shows	Lesson
Mean	Centre	Mean, Median and Mode
Standard deviation	Spread	Variance and Standard Deviation
Skewness	Asymmetry	Skewness and Kurtosis
Kurtosis	Tail heaviness	Skewness and Kurtosis
Quantiles (percentiles)	Values at chosen probabilities, such as the worst 5%	Value at Risk (VaR)

The cumulative distribution function#

The cumulative distribution function (CDF) gives the probability that a value is at or below a level:

F(x) = P(X ≤ x)

Value at risk uses the CDF: the 5% VaR is the loss level such that losses are worse only 5% of the time. See Value at Risk (VaR).

Choosing a distribution#

Fitting distributions to data#

Plot a histogram and compare it with candidate distributions.
Use a Q Q plot to check tails against a theoretical distribution.
Estimate parameters, often by maximum likelihood. See Maximum Likelihood.
Test goodness of fit (such as Kolmogorov Smirnov or Jarque Bera tests for normality).
Check stability over time; distributions change across regimes. See Structural Breaks and Regime Changes.

Empirical distributions#

Instead of assuming a formula, traders can use the actual historical distribution of returns, for example in historical simulation VaR or bootstrapping. This captures real features like fat tails, but is limited to what happened in the sample. See Bootstrap and Permutation Tests.

Frequently asked questions#

What is a probability distribution?#

A description of all possible values of an uncertain quantity and the probability of each value or range of values.

Are stock returns normally distributed?#

Not exactly. Returns have fatter tails and more extreme moves than the normal distribution predicts, though the normal can be a rough approximation.

Which distributions matter most in trading?#

The normal, lognormal, Student's t, binomial and Poisson distributions, along with fat tailed and mixture distributions for risk.

Next, study the most famous distribution in Normal Distribution.

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Next lessonNormal DistributionThe normal distribution is the bell curve behind many financial models. Learn its properties, the 68 95 99.7 rule, where traders use it and why markets break it.