A learning path from uncertainty to distribution choice
A probability distribution connects possible outcomes with their likelihoods. Begin with events and conditional probability, distinguish counts from continuous measurements, then learn how cumulative probabilities and densities work. The remaining articles explain six distribution families through assumptions, numbers and Excel.
Each article includes worked illustrations and questions with expandable answers. Distribution articles also include a calculator and a chart. You can print any article or save it as a PDF using your browser.
1. Basics of Probability
Events, conditional probability, independence and Bayes’ theorem, explained through finance examples.
2. Discrete vs Continuous Probability Distributions
Countable outcomes, continuous measurements, probability mass and probability density.
3. Cumulative Distribution Function vs Probability Density Function
Cumulative probabilities, densities, probability masses and quantiles.
4. Normal and Standard Normal Distributions
The bell curve, standardisation, tail probabilities and finance applications.
5. Lognormal Distribution
Positive values, multiplicative growth and the distinction between log parameters and level parameters.
6. Student’s t-Distribution
Heavier tails, degrees of freedom, inference and the difference between scale and standard deviation.
7. Binomial Distribution
Repeated binary trials, default counts and exact versus cumulative probabilities.
8. Triangular Distribution
A bounded model using minimum, most likely and maximum estimates.
9. Beta Distribution
Flexible shapes on a bounded interval, proportions and learning about an unknown probability.
Which distribution matches the variable?
| Variable or task | Candidate model | Main assumption to inspect |
|---|---|---|
| An unrestricted symmetric measurement | Normal | Tail behaviour and stability |
| A strictly positive value with multiplicative growth | Lognormal | Normality of the logarithm |
| Heavy-tailed symmetric observations or a mean statistic | Student’s t | Degrees of freedom, scale and sampling assumptions |
| Success count in a fixed set of trials | Binomial | Independent trials with a common probability |
| An uncertain value with estimated bounds and a mode | Triangular | Credibility of hard bounds and the assumed shape |
| A proportion between zero and one | Beta | Shape, bounds and endpoint masses |
A common notation guide
| Symbol or abbreviation | Meaning |
|---|---|
| P(A) | Probability of event A |
| X | Random variable |
| μ and σ | Mu: mean or log-mean as specified. Sigma: standard deviation or scale as specified. |
| CDF | Cumulative distribution function, F(x) = P(X ≤ x) |
| Probability density function for an absolutely continuous variable | |
| PMF | Probability mass function for a discrete variable |
| ln and exp | Natural logarithm and exponential function |