Common Continuous Distributions
From bell curves to waiting times. Six families that model the measured world.
Probability is now an Area
For continuous variables, P(X = x) = 0. Probability lives in intervals, calculated as the area under the PDF.
Gamma → Beta → Log-normal
Uniform: Equal Likelihood
Flat PDF over [a, b]. The most uninformative distribution when you only know the range.
Normal: The Bell Curve
Symmetric, parameterized by μ (center) and σ (spread). Appears everywhere due to the Central Limit Theorem.
68 – 95 – 99.7
- 68% of values within μ ± 1σ
- 95% of values within μ ± 2σ
- 99.7% of values within μ ± 3σ
a table or calculator
Standard Normal & Z-Scores
Z ~ N(0,1). Every normal X converts to Z: measures standard deviations from the mean.
Exponential: Waiting Time
Time until the first Poisson event. Memoryless: past waiting time tells you nothing about future waiting time.
Exponential CDF
Gamma: k-th Event
Waiting time for the k-th Poisson event. Shape k, rate λ. Sum of k independent Exponential(λ) variables.
Beta: Proportions
Lives on [0, 1]. The natural model for proportions, success rates, and Bayesian priors. Beta(1,1) = Uniform.
After k successes in n:
Posterior: Beta(α+k, β+n−k)
Choosing the Right Distribution
- Bounded interval, no preference → Uniform
- Bell-shaped, sum of influences → Normal
- Waiting for 1st event → Exponential
- Waiting for k-th event → Gamma
- Proportion in [0,1] → Beta
- Positive, right-skewed → Log-normal
What you learned
Six continuous distributions, six stories. The Normal is the most important — and the Central Limit Theorem explains why it's everywhere.