Statistical distributions are mathematical functions that describe the probability of different outcomes in a random process, forming the foundation of probability theory, statistical inference, and data analysis across all quantitative fields. Most foundational distributions belong to the exponential family, a unifying mathematical framework that guarantees sufficient statistics and tractable inference. Each distribution is characterized by its parameters (shape, location, scale), and choosing the correct distribution for your data determines the validity of subsequent statistical tests, predictions, and decisions. A key distinction to always keep in mind: discrete distributions model countable outcomes (coin flips, customer arrivals), while continuous distributions model measurable quantities (temperature, time, income)—confusing the two leads to fundamentally incorrect analysis.
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This topic spans 17 focused tables and 152 indexed concepts, 133 flashcards. Below is a complete table-by-table outline of this topic, spanning foundational concepts through advanced details.
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Table 1: Core Discrete Distributions
These model outcomes you can count—heads or tails, customer arrivals, defective items in a batch. The whole family branches out from a single Bernoulli trial: repeat it a fixed number of times and you get the binomial, count failures until a success and you get the geometric, and so on through the rarer cousins that handle finite populations, unequal probabilities, or overdispersed counts.
| Distribution | Example | Description | |
|---|---|---|---|
p = 0.3P(X=1) = 0.3, P(X=0) = 0.7 | • Models a single trial with two outcomes (success/failure) • parameter p is success probability• mean = p, variance = p(1−p) | ||
n=10, p=0.3$P(X{=}k) = \binom{10}{k} 0.3^k \cdot 0.7^{10-k}$ | • Number of successes in n independent Bernoulli trials • models fixed trials with replacement • mean = np, variance = np(1−p) | ||
λ=5$P(X{=}k) = \frac{5^k e^{-5}}{k!}$ | • Counts events in a fixed interval (time/space) when events occur independently at constant rate λ • mean = variance = λ | ||
p=0.2$P(X{=}k) = 0.8^{k-1} \cdot 0.2$ | • Number of trials until first success in independent Bernoulli trials • memoryless discrete distribution • mean = 1/p | ||
r=3, p=0.4$P(X{=}k) = \binom{k+r-1}{k} 0.4^r \cdot 0.6^k$ | • Number of failures before r successes • generalizes geometric (r=1) • models overdispersed count data (variance > mean) | ||
n=10, p=[0.3,0.5,0.2]$P(\mathbf{x}) = \frac{10!}{x_1! x_2! x_3!} 0.3^{x_1} 0.5^{x_2} 0.2^{x_3}$ | • Generalizes binomial to k>2 mutually exclusive outcomes • models categorical trials • sum of counts = n | ||
N=50, K=20, n=10$P(X{=}k) = \frac{\binom{20}{k}\binom{30}{10-k}}{\binom{50}{10}}$ | • Successes in sampling without replacement from finite population • use when sample size > 5% of population • binomial analog for finite populations | ||
a=1, b=6P(X=k) = 1/6 for k∈{1,2,3,4,5,6} | • Equal probability for each of n outcomes • models fair dice, random selection • mean = (a+b)/2 | ||
p=[0.2,0.3,0.5]P(X=k) = pₖ | • Single trial with k>2 possible outcomes • multinomial with n=1 • generalizes Bernoulli to multiple categories | ||
n=10, α=2, β=5p ~ Beta(α,β), X ~ Bin(n,p) | • Overdispersed binomial where success probability itself is random (Beta-distributed) • models heterogeneous populations • used in A/B testing, ecology | ||
$P(X{=}k) = \frac{1/k^s}{\sum_{n=1}^{N} 1/n^s}$ | • Power-law over ranked items • frequency inversely proportional to rank • models word frequencies, city sizes, web traffic | ||
pᵢ = [0.1, 0.4, 0.7, 0.9]X = Σ Bernoulli(pᵢ) | • Sum of independent Bernoulli trials with unequal success probabilities • generalizes binomial (all pᵢ equal) • used in voting models, reliability | ||
μ₁=3, μ₂=2X = N₁ − N₂, Nᵢ ~ Poisson(μᵢ) | • Distribution of difference of two independent Poissons • can be negative • used in sports analytics (goal/point differentials) |