02_distributions.rst

Tutorial 2: Distributions Guide

Complete guide to all 30 distributions.

Continuous Distributions

Normal Distribution

Use when: Data is symmetric, bell-shaped, no outliers

from distfit_pro import get_distribution
import numpy as np

# Heights of adult males (cm)
data = np.random.normal(175, 7, 1000)

dist = get_distribution('normal')
dist.fit(data)
print(dist.summary())

Parameters: - loc (μ): mean/center - scale (σ): standard deviation

When NOT to use: - Skewed data - Heavy tails - Bounded data (e.g., percentages)

Lognormal Distribution

Use when: Data is positive, right-skewed

# Income data ($)
data = np.random.lognormal(10, 0.5, 1000)

dist = get_distribution('lognormal')
dist.fit(data)
print(dist.summary())

Parameters: - s (σ): shape (log-scale std) - scale (exp(μ)): scale

Common applications: - Income/wealth - Stock prices - File sizes - Particle sizes

Weibull Distribution

Use when: Modeling time-to-failure, lifetimes

# Component lifetime (hours)
data = np.random.weibull(1.5, 1000) * 1000

dist = get_distribution('weibull')
dist.fit(data)
print(dist.summary())

# Reliability at t=500 hours
reliability = dist.reliability(500)
print(f"Reliability at 500h: {reliability:.4f}")

Parameters: - c (k): shape

• k < 1: decreasing failure rate (infant mortality)
• k = 1: constant failure rate (random failures)
• k > 1: increasing failure rate (wear-out)

• scale (λ): scale

Applications: - Reliability engineering - Failure time analysis - Wind speed modeling

Gamma Distribution

Use when: Waiting times, sum of exponentials

# Waiting time for 5 events
data = np.random.gamma(5, 2, 1000)

dist = get_distribution('gamma')
dist.fit(data)
print(dist.summary())

Parameters: - a (α): shape - scale (θ): scale

Special cases: - α = 1: Exponential distribution - α = k/2, θ = 2: Chi-square with k df

Exponential Distribution

Use when: Time between events (memoryless)

# Time between arrivals (minutes)
data = np.random.exponential(5, 1000)

dist = get_distribution('exponential')
dist.fit(data)
print(dist.summary())

# Probability of waiting < 3 minutes
prob = dist.cdf(np.array([3]))[0]
print(f"P(wait < 3 min) = {prob:.4f}")

Key property: Memoryless!

# P(X > 10 | X > 5) = P(X > 5)
# Past doesn't affect future

Beta Distribution

Use when: Data is bounded between 0 and 1

# Success rates, percentages
data = np.random.beta(2, 5, 1000)

dist = get_distribution('beta')
dist.fit(data)
print(dist.summary())

Parameters: - a (α): shape 1 - b (β): shape 2

Applications: - Conversion rates - Probabilities - Proportions - Bayesian priors

Pareto Distribution

Use when: Power-law, heavy tails, 80-20 rule

# Wealth distribution
data = (np.random.pareto(2, 1000) + 1) * 50000

dist = get_distribution('pareto')
dist.fit(data)
print(dist.summary())

Applications: - Wealth/income distribution - City sizes - Word frequencies

Student's t Distribution

Use when: Small samples, heavier tails than normal

# Small sample data
data = np.random.standard_t(5, 100)

dist = get_distribution('studentt')
dist.fit(data)
print(dist.summary())

Parameters: - df (ν): degrees of freedom - As df → ∞, approaches Normal

Discrete Distributions

Poisson Distribution

Use when: Count of rare events in fixed interval

# Number of calls per hour
data = np.random.poisson(lam=3.5, size=1000)

dist = get_distribution('poisson')
dist.fit(data)
print(dist.summary())

# P(exactly 5 calls)
prob = dist.pdf(np.array([5]))[0]
print(f"P(X = 5) = {prob:.4f}")

Parameter: - mu (λ): rate (mean = variance)

Applications: - Call center arrivals - Website visitors - Defects in manufacturing

Binomial Distribution

Use when: n independent yes/no trials

# 10 coin flips, p=0.5
data = np.random.binomial(n=10, p=0.5, size=1000)

dist = get_distribution('binomial')
dist.fit(data)
print(dist.summary())

Parameters: - n: number of trials - p: success probability

Applications: - Quality control (pass/fail) - Survey responses (yes/no) - A/B testing

Negative Binomial

Use when: Overdispersed count data (variance > mean)

# Overdispersed counts
data = np.random.negative_binomial(5, 0.5, 1000)

dist = get_distribution('nbinom')
dist.fit(data)
print(dist.summary())

Better than Poisson when: - Data shows more variability - Clustering of events

Complete Distribution List

Continuous (25):

1. Normal - symmetric, bell curve
2. Lognormal - positive, right-skewed
3. Weibull - reliability, lifetimes
4. Gamma - waiting times
5. Exponential - time between events
6. Beta - bounded [0,1]
7. Uniform - equal probability
8. Triangular - three-point estimate
9. Logistic - growth models
10. Gumbel - extreme values (max)
11. Frechet - extreme values (positive)
12. Pareto - power law, 80-20
13. Cauchy - undefined mean/variance
14. Student's t - heavy tails
15. Chi-squared - variance tests
16. F - variance ratio
17. Rayleigh - signal processing
18. Laplace - sparse data
19. Inverse Gamma - Bayesian priors
20. Log-Logistic - survival analysis

Discrete (5):

1. Poisson - rare event counts
2. Binomial - n trials
3. Negative Binomial - overdispersed
4. Geometric - trials to first success
5. Hypergeometric - sampling without replacement

Choosing the Right Distribution

Decision Tree:

1. Is data discrete (counts) or continuous?
   • Discrete → Poisson, Binomial, etc.
   • Continuous → continue
2. Is data bounded?
   • [0, 1] → Beta
   • [a, b] → Uniform, Triangular
   • [0, ∞) → Lognormal, Gamma, Weibull, Exponential
   • (-∞, ∞) → Normal, Logistic, Cauchy, Student's t
3. Is data skewed?
   • Right-skewed → Lognormal, Gamma, Weibull
   • Symmetric → Normal, Logistic, Student's t
   • Left-skewed → Reflected versions
4. Heavy tails?
   • Yes → Student's t, Cauchy, Pareto
   • No → Normal, Logistic
5. Special domain?
   • Reliability → Weibull, Exponential
   • Extreme values → Gumbel, Frechet
   • Survival → Weibull, Log-Logistic

Next Steps

• :doc:`03_fitting_methods` - Different ways to fit
• :doc:`04_gof_tests` - Test if fit is good