Probability Distributions Cheat Sheet for Finance
A probability distribution describes how the possible values of a random variable are spread out. In finance, distributions model everything from stock returns and default probabilities to option pricing and portfolio risk. This cheat sheet covers the distributions you will actually use on the job and on the CFA exam.
Why Distributions Matter in Finance
Every financial model makes assumptions about how data behaves. When you run a regression, price an option using Black-Scholes, or calculate standard deviation for a portfolio, you are relying on a specific distribution. Pick the wrong one and your risk estimates, valuations, and confidence intervals fall apart.
Distributions also drive volatility modeling, stress testing, and Monte Carlo simulations. Understanding their properties is not optional — it is foundational.
Discrete vs. Continuous Distributions
| Feature | Discrete | Continuous |
|---|---|---|
| Outcomes | Countable (0, 1, 2, …) | Any value in a range |
| Probability | P(X = x) can be > 0 | P(X = x) = 0; use intervals |
| Function | Probability Mass Function (PMF) | Probability Density Function (PDF) |
| Finance example | Number of defaults in a portfolio | Stock returns over time |
Key Discrete Distributions
Bernoulli Distribution
A single trial with two outcomes: success (p) or failure (1 − p). Think of it as a coin flip. In finance, a single bond either defaults or it does not.
Binomial Distribution
The number of successes in n independent Bernoulli trials. Use it when you want to know: out of 100 corporate bonds, how many might default?
Mean: np | Variance: np(1 − p)
Poisson Distribution
Models the count of events in a fixed interval when events occur independently at a constant average rate (λ). Common for modeling trade arrivals, insurance claims, or rare default events.
Mean = Variance = λ. When λ is large, the Poisson approaches a normal distribution.
Key Continuous Distributions
Uniform Distribution
Every outcome in the range [a, b] is equally likely. Used in Monte Carlo simulations as the starting point for generating random variables.
Normal (Gaussian) Distribution
The bell curve. Defined by mean (μ) and standard deviation (σ). Most financial models assume returns are normally distributed — which works reasonably well for large, diversified portfolios but underestimates tail risk.
| Property | Value |
|---|---|
| Mean | μ |
| Variance | σ² |
| Skewness | 0 (symmetric) |
| Kurtosis | 3 (mesokurtic) |
| Key use | Sharpe ratio, beta, VaR (parametric) |
Standard Normal Distribution (Z)
A normal distribution with μ = 0 and σ = 1. You convert any normal variable to Z using: Z = (X − μ) / σ. Z-tables give you cumulative probabilities, which are essential for hypothesis testing and confidence intervals.
Lognormal Distribution
If ln(X) is normally distributed, then X is lognormally distributed. Stock prices are modeled as lognormal because prices cannot go below zero, but returns can be modeled as normal. This is the assumption behind Black-Scholes option pricing.
Student’s t-Distribution
Looks like a normal distribution but with fatter tails. Used when sample sizes are small or population variance is unknown. As degrees of freedom increase, it converges to the normal. Critical for regression analysis and hypothesis tests in finance.
Chi-Square Distribution
The sum of squared standard normal variables. Used for variance testing. If you want to test whether a portfolio’s volatility has changed, this is the distribution you use.
F-Distribution
The ratio of two chi-square variables divided by their degrees of freedom. Used in ANOVA and for testing whether two variances are equal — for instance, comparing volatility across two time periods.
Distribution Comparison Table
| Distribution | Type | Parameters | Finance Application |
|---|---|---|---|
| Bernoulli | Discrete | p | Default / no default |
| Binomial | Discrete | n, p | Number of defaults in portfolio |
| Poisson | Discrete | λ | Trade arrivals, rare events |
| Uniform | Continuous | a, b | Monte Carlo seed values |
| Normal | Continuous | μ, σ | Returns, VaR, Sharpe ratio |
| Lognormal | Continuous | μ, σ | Stock prices, Black-Scholes |
| Student’s t | Continuous | df | Small-sample tests, regression |
| Chi-Square | Continuous | df | Variance testing |
| F | Continuous | df₁, df₂ | Comparing variances, ANOVA |
Skewness and Kurtosis
Skewness measures asymmetry. Equity returns typically show negative skew — large drops are more common than large gains of the same magnitude. Kurtosis measures tail thickness. Financial returns exhibit excess kurtosis (leptokurtic), meaning extreme events happen more often than a normal distribution predicts. This is why VaR models that assume normality underestimate risk.
| Measure | Normal Value | Typical Equity Returns | Implication |
|---|---|---|---|
| Skewness | 0 | Negative | More large losses than gains |
| Kurtosis | 3 | > 3 (leptokurtic) | Fat tails — more extreme events |
Never assume normality without checking. Run a Jarque-Bera test on your data. If skewness ≠ 0 or kurtosis ≠ 3, your confidence intervals and VaR estimates based on the normal distribution will be wrong. Consider using the Student’s t or a non-parametric approach instead.
Confusing the normal distribution (for returns) with the lognormal distribution (for prices). Stock returns can be negative — which is fine for a normal distribution. But stock prices cannot be negative — which is why prices follow a lognormal. Mixing them up leads to nonsensical model outputs.
Key Takeaways
- Discrete distributions (binomial, Poisson) count events; continuous distributions (normal, lognormal) model values in a range.
- The normal distribution is the backbone of most financial models, but it underestimates tail risk.
- Stock prices follow a lognormal distribution; stock returns follow a normal distribution.
- Use the Student’s t distribution when sample sizes are small or variance is unknown.
- Always check skewness and kurtosis — real financial data rarely behaves like a textbook bell curve.
Frequently Asked Questions
What probability distribution do stock returns follow?
Stock returns are commonly modeled as normally distributed, especially over short periods. However, empirical data shows that real returns have fatter tails and slight negative skew, meaning extreme events occur more often than the normal distribution predicts.
Why do stock prices follow a lognormal distribution?
Because stock prices cannot go below zero. If you model continuously compounded returns as normal, the resulting price levels will be lognormally distributed. This is the key assumption behind the Black-Scholes option pricing model.
What is the difference between the normal and t-distribution?
The t-distribution has heavier tails than the normal, which accounts for additional uncertainty when the sample size is small or population variance is unknown. As degrees of freedom increase (roughly above 30), the t-distribution converges to the normal.
When should I use the Poisson distribution in finance?
Use Poisson when modeling the count of rare, independent events in a fixed time interval — such as the number of bond defaults per year, insurance claims per month, or trade executions per minute.
What are fat tails and why do they matter?
Fat tails (leptokurtosis) mean extreme outcomes occur more frequently than a normal distribution predicts. In finance, this is critical because models assuming normality will underestimate the probability of market crashes, leading to inadequate risk management and potential large losses.