Statistical Distributions
Key Discrete Distributions
| Property | Statement |
|---|---|
| Binomial | \(B(n,p)\): n trials, success prob p |
| Poisson | \(\text{{Pois}}(\lambda)\): rare events, mean λ |
| Geometric | Trials until first success |
Key Continuous Distributions
| Property | Statement |
|---|---|
| Normal | \(N(\mu,\sigma^2)\): bell curve |
| Exponential | Time between Poisson events |
| Uniform | \(U(a,b)\): equal density on [a,b] |
| t | Heavy-tailed; used when σ unknown |
| Chi-squared | Sum of squared normals; used in tests |
Examples
Example 1. X~Poisson(3). Find P(X=2).
Solution. P(X=2) = e⁻³·3²/2! = 9e⁻³/2 ≈ 0.224.
Background
Statistical distributions are mathematical models for random phenomena. Choosing the right distribution for a problem requires understanding the data-generating process: count data suggest Poisson or binomial; waiting times suggest exponential or gamma; proportions suggest beta; extreme values suggest Gumbel or Pareto.
The binomial distribution \(B(n,p)\) models the number of successes in \(n\) independent Bernoulli trials. Mean \(np\), variance \(np(1-p)\). For large \(n\) and moderate \(p\), it is approximated by \(N(np, np(1-p))\); for large \(n\) and small \(p\), by Poisson(\(np\)).
The Poisson distribution models rare events in a fixed interval of time or space. If events occur at rate \(\lambda\), the number in a unit interval is Poisson(\(\lambda\)). The exponential distribution models the waiting time between Poisson events.
The t-distribution with \(\nu\) degrees of freedom has heavier tails than the normal. As \(\nu\to\infty\), it converges to \(N(0,1)\). The chi-squared distribution \(\chi^2(\nu)\) is the sum of \(\nu\) squared standard normals; it arises in variance estimation and goodness-of-fit tests. The F-distribution is the ratio of two chi-squared variables divided by their degrees of freedom; it appears in ANOVA and regression.
Further Reading & Context
The study of statistical distributions connects to many areas of mathematics and its applications. Understanding the foundational definitions and theorems provides the basis for advanced work in analysis, algebra, and applied mathematics.
Historical development: most mathematical concepts evolved over centuries, with contributions from mathematicians across many cultures. The modern axiomatic treatment provides rigor, while computational tools enable practical application.
In modern mathematics, this topic appears in graduate courses and research across pure and applied mathematics. Connections to computer science, physics, and engineering make it a versatile and important area of study. Mastery of the core results and techniques opens doors to research in number theory, analysis, geometry, and beyond.
Recommended next steps: work through the standard theorems with full proofs, explore the connections to related topics listed above, and practice with a variety of problems ranging from computational exercises to theoretical proofs. The interplay between different areas of mathematics is one of the subject's greatest rewards.