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Probability & Distributions

Model uncertainty with probability and understand how random variables behave.

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Probability Basics

P(A) = favorable outcomes / total outcomes
0 ≤ P(A) ≤ 1
P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
P(A') = 1 − P(A)

Probability quantifies uncertainty. These rules come from set theory — unions and intersections. Counting techniques like combinatorics (permutations and combinations) are essential for computing probabilities in finite sample spaces.

Conditional Probability & Bayes' Theorem

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P(A|B) = P(A ∩ B)/P(B)
Independent events: P(A ∩ B) = P(A)·P(B)
Bayes: P(A|B) = P(B|A)·P(A)/P(B)

Bayes' Theorem is the backbone of Bayesian inference and machine learning. It lets us update beliefs with new evidence. Total probability connects this to partition: P(B) = Σ P(B|Aᵢ)·P(Aᵢ).

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Example: Disease Testing

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Disease prevalence: 1%. Test sensitivity: 99%. False positive rate: 5%.

P(Disease | Positive) = (0.99×0.01)/(0.99×0.01 + 0.05×0.99) ≈ 0.0099/0.0594 ≈ 16.7%

Even with a good test, a positive result is only 16.7% likely to be a true positive when the disease is rare!

Discrete Distributions

  • Bernoulli: Single trial, p = success. E[X] = p, Var = p(1−p)
  • Binomial(n, p): P(X = k) = C(n,k)·pᵏ(1−p)ⁿ⁻ᵏ — uses polynomial expansion
  • Poisson(λ): P(X = k) = e⁻λ·λᵏ/k! — models rare events per interval
  • Geometric(p): P(X = k) = (1−p)ᵏ⁻¹·p — trials until first success. Connects to exponential functions

Continuous Distributions

Normal: f(x) = (1/σ√(2π))·e^(−(x−μ)²/(2σ²))
Z-score: z = (x − μ)/σ
Standard normal: μ = 0, σ = 1

For continuous distributions, probabilities are areas under the curve — you need integration. The normal (Gaussian) distribution is the most important, governing everything from measurement error to stock prices.

  • Uniform(a, b): f(x) = 1/(b−a) — constant density
  • Exponential(λ): f(x) = λe⁻λˣ — time between events
  • t-distribution: Used in hypothesis testing with small samples
  • Chi-squared: Sum of squared standard normals — used in goodness-of-fit tests

Central Limit Theorem

Central Limit Theorem: Regardless of the population distribution, the sample mean X̄ approaches a normal distribution N(μ, σ²/n) as n → ∞. This is why the normal distribution is so important, and why statistical inference works. The convergence concept mirrors limits in calculus.

Related Topics

Descriptive StatisticsSummarize data before building probability models. IntegralsContinuous probabilities require integration under density curves. Number TheoryCombinatorics and counting underpin discrete probability. FunctionsExponential and logarithmic functions appear throughout probability.