# 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.

Source: https://learn.tradelabsai.com/math/probability-distributions/  
Track: Math and Statistics · Level: Advanced · Updated: 2026-10-03  
Publisher: TradeLabs AI (https://tradelabsai.com). Education, not financial advice.  
Cite as: TradeLabs Learn, "Probability Distributions Explained", https://learn.tradelabsai.com/math/probability-distributions/

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](https://learn.tradelabsai.com/math/normal-distribution/) |
| Lognormal | Prices that cannot go below zero; Black Scholes | [Lognormal Distribution](https://learn.tradelabsai.com/math/lognormal-distribution/) |
| Student's t | Fat tailed returns; small sample tests | [Student's t-Distribution](https://learn.tradelabsai.com/math/students-t-distribution/) |
| Binomial | Number of wins in a fixed number of trades | [Binomial and Bernoulli Distributions](https://learn.tradelabsai.com/math/binomial-distribution/) |
| Poisson | Number of rare events in a period | [Poisson and Exponential Distributions](https://learn.tradelabsai.com/math/poisson-distribution/) |
| Bernoulli | Single yes or no outcome, such as a prediction market | [Random Variables](https://learn.tradelabsai.com/math/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](https://learn.tradelabsai.com/math/mixture-distributions/) |
| Power laws and extreme value distributions | The largest losses in the tails | [Fat Tails](https://learn.tradelabsai.com/math/fat-tails/) |

## Describing a distribution

| Measure | What it shows | Lesson |
|---|---|---|
| Mean | Centre | [Mean, Median and Mode](https://learn.tradelabsai.com/math/mean-median-and-mode/) |
| Standard deviation | Spread | [Variance and Standard Deviation](https://learn.tradelabsai.com/math/variance-and-standard-deviation/) |
| Skewness | Asymmetry | [Skewness and Kurtosis](https://learn.tradelabsai.com/math/skewness-and-kurtosis/) |
| Kurtosis | Tail heaviness | [Skewness and Kurtosis](https://learn.tradelabsai.com/math/skewness-and-kurtosis/) |
| Quantiles (percentiles) | Values at chosen probabilities, such as the worst 5% | [Value at Risk (VaR)](https://learn.tradelabsai.com/portfolio/value-at-risk/) |

## 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)](https://learn.tradelabsai.com/portfolio/value-at-risk/).

## Choosing a distribution

**Example: Normal vs fat tailed assumptions**
A portfolio's daily returns have a standard deviation of 1%. Under a normal distribution, a daily loss worse than 4% (a 4 standard deviation move) should happen about once every 120 years of trading days. Under a Student's t distribution with 4 degrees of freedom, scaled to the same standard deviation, it happens roughly once every one to two years. Real equity markets have experienced 4 standard deviation days far more often than the normal model implies. The choice of distribution changes the risk picture completely. See [Fat Tails](https://learn.tradelabsai.com/math/fat-tails/).

## Fitting distributions to data

1. **Plot a histogram** and compare it with candidate distributions.
2. **Use a Q Q plot** to check tails against a theoretical distribution.
3. **Estimate parameters,** often by maximum likelihood. See [Maximum Likelihood](https://learn.tradelabsai.com/math/maximum-likelihood/).
4. **Test goodness of fit** (such as Kolmogorov Smirnov or Jarque Bera tests for normality).
5. **Check stability** over time; distributions change across regimes. See [Structural Breaks and Regime Changes](https://learn.tradelabsai.com/math/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](https://learn.tradelabsai.com/math/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](https://learn.tradelabsai.com/math/normal-distribution/).

## Continue learning

- Next lesson: [Normal Distribution](https://learn.tradelabsai.com/math/normal-distribution/)
- Previous lesson: [Bayesian Statistics](https://learn.tradelabsai.com/math/bayesian-statistics/)
- Related: [Bayesian Statistics](https://learn.tradelabsai.com/math/bayesian-statistics/): Bayesian statistics combines prior beliefs with data to estimate uncertain quantities. Learn priors and posteriors, shrinkage, credible intervals and trading uses.
- Related: [Normal Distribution](https://learn.tradelabsai.com/math/normal-distribution/): The 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.
- Related: [Lognormal Distribution](https://learn.tradelabsai.com/math/lognormal-distribution/): A lognormal distribution describes values whose log is normal, like prices that cannot go negative. Learn log returns, volatility drag and its option uses.
- Related: [Student's t-Distribution](https://learn.tradelabsai.com/math/students-t-distribution/): The Student's t distribution has fatter tails than the normal. Learn how it is used for small sample tests and to model fat tailed returns, with examples.
- Related: [Fat Tails](https://learn.tradelabsai.com/math/fat-tails/): Fat tails mean extreme market moves happen far more often than the normal curve predicts. Learn the evidence, the causes, how to measure them and how to manage them.
- Related: [Random Variables](https://learn.tradelabsai.com/math/random-variables/): A random variable assigns numbers to uncertain outcomes, like a trade's profit. Learn discrete and continuous variables, expectation and variance, with examples.