calculate-probability-distribution

name	calculate-probability-distribution
description	Use when working with probability distributions — identifying the correct distribution for a phenomenon, computing probabilities and quantiles, fitting distributions to data, and checking distributional assumptions statistically.
source	DeGroot & Schervish "Probability and Statistics" 4th ed. (2012); Casella & Berger "Statistical Inference" 2nd ed. (2002); Johnson et al. "Univariate Discrete Distributions" 3rd ed. (2005)
tags	["probability","statistics","distribution-fitting","hypothesis-testing","Bayesian","quantiles","random-variables"]

Calculate Probability Distribution

Identify and apply the correct probability distribution for a phenomenon — computing probabilities, fitting parameters from data, and validating distributional assumptions with goodness-of-fit tests — to support reliable probabilistic inference and simulation.

Why This Is Best Practice

Adopted by: Actuarial science (loss distributions), reliability engineering (lifetime distributions), queueing theory (arrival processes), machine learning (likelihood functions), Bayesian inference (prior/posterior), and risk management all depend on correct probability distribution selection. SOA (Society of Actuaries) and CAS (Casualty Actuarial Society) exams test distribution fitting as a core competency. SciPy, R, and MATLAB provide standardized implementations of >100 distributions. Impact: Misidentifying the distribution for a phenomenon leads to incorrect probability estimates. DeGroot & Schervish (2012) demonstrate that the difference between normal and heavy-tailed (e.g., Cauchy) distribution assumptions produces probability estimates that differ by orders of magnitude in the tails — precisely where risk decisions are made. Fitting a normal distribution to financial returns (which are leptokurtic) underestimates tail risk (Black Swan events) catastrophically.

Steps

1. Match distribution type to the phenomenon

Select distribution family based on the variable's nature:

Discrete distributions:

Phenomenon	Distribution	Parameters
Binary outcome (success/failure)	Bernoulli	p
Count of successes in n trials	Binomial	n, p
Count until first success	Geometric	p
Count of rare events in interval	Poisson	λ
Count of successes without replacement	Hypergeometric	N, K, n

Continuous distributions:

Phenomenon	Distribution	Parameters
Symmetric bell-shaped (Central Limit Theorem)	Normal (Gaussian)	μ, σ
Positively skewed, right-tailed	Log-normal, Gamma, Weibull	—
Uniform random selection	Uniform	a, b
Time between Poisson events	Exponential	λ
Lifetime / reliability modeling	Weibull	k, λ
Heavy-tailed extreme values	Pareto, GEV	α, x_min
Proportions (bounded 0,1)	Beta	α, β

2. Compute key probabilities and quantiles

For a Normal distribution X ~ N(μ, σ²):

from scipy import stats
dist = stats.norm(loc=mu, scale=sigma)
prob = dist.cdf(x)           # P(X ≤ x)
prob_range = dist.cdf(b) - dist.cdf(a)  # P(a ≤ X ≤ b)
x_q = dist.ppf(q)            # quantile: P(X ≤ x_q) = q

For discrete distributions (Binomial):

dist = stats.binom(n=n, p=p)
prob_exact = dist.pmf(k)     # P(X = k)
prob_at_most = dist.cdf(k)   # P(X ≤ k)

Key quantiles:

• q = 0.95 → 95th percentile (one-sided 5% significance threshold)
• q = 0.975 → 97.5th percentile (two-sided 5% significance: ±1.96σ for Normal)
• q = 0.999 → 99.9th percentile (1-in-1000 event)

3. Fit distribution parameters from data

Maximum Likelihood Estimation (MLE) — the standard approach:

import numpy as np
from scipy import stats

data = np.array([...])

# Fit normal distribution (returns mu_hat, sigma_hat)
mu_hat, sigma_hat = stats.norm.fit(data)

# Fit exponential distribution
loc_hat, scale_hat = stats.expon.fit(data, floc=0)  # fix location at 0
lambda_hat = 1 / scale_hat  # rate parameter

# Compare multiple candidate distributions
for dist_name in ['norm', 'lognorm', 'expon', 'gamma', 'weibull_min']:
    dist = getattr(stats, dist_name)
    params = dist.fit(data)
    ks_stat, ks_p = stats.kstest(data, dist_name, args=params)
    print(f"{dist_name}: KS p-value = {ks_p:.4f}")

Method of Moments: equate sample moments (mean, variance) to theoretical moments; less efficient than MLE but useful as starting value.

4. Validate distributional assumptions

Always verify fit before using in downstream analysis:

Graphical tests:

• Q-Q plot: plot sample quantiles vs. theoretical quantiles; points on the diagonal line = good fit; systematic deviations indicate wrong distribution family
• Histogram + fitted PDF overlay: visual assessment of fit

Statistical goodness-of-fit tests:

• Kolmogorov-Smirnov (KS) test: compares empirical vs. theoretical CDF; non-parametric; sensitive to the center of distribution
• Anderson-Darling test: like KS but more sensitive to tails; preferred for reliability/risk applications
• Chi-squared goodness-of-fit: for discrete distributions; requires adequate counts per bin (expected > 5)

P-value > 0.05 → insufficient evidence to reject the distribution (not proof it is correct).

5. Work with joint and conditional distributions

For two random variables X, Y:

Joint: f(x, y) = probability density at (x, y)
Marginal: f_X(x) = ∫ f(x, y) dy
Conditional: f(y|x) = f(x, y) / f_X(x)
Independence: f(x, y) = f_X(x) × f_Y(y)

Covariance and correlation:

Cov(X, Y) = E[(X-μX)(Y-μY)]
ρ = Cov(X,Y) / (σX σY) ∈ [−1, +1]

6. Apply the Central Limit Theorem where appropriate

For sample mean X̄ of n i.i.d. random variables with mean μ and variance σ²:

X̄ ~ N(μ, σ²/n)  approximately, for large n (typically n ≥ 30)

This justifies normal-approximation methods and confidence intervals for population means regardless of the underlying distribution.

Common Mistakes

• Assuming normality without checking: Test with Q-Q plot and Anderson-Darling before using normal-theory inference. Heavy-tailed data (financial, insurance claims) produces intervals and p-values that are systematically wrong under normality.
• Fitting many distributions and choosing the best-fitting without correction: Multiple testing inflates false discovery rate. Use AIC/BIC for model selection when fitting many distributions to the same data.
• Confusing the Poisson and Binomial for large n: For n large and p small, Binomial(n,p) ≈ Poisson(np). The approximation holds well when n > 100 and np < 10.

When NOT to Use

• Multivariate data with complex dependencies: use copulas or multivariate distributions (MVN, Dirichlet) rather than independent univariate fits — marginal fit correctness does not imply joint distribution fit.