https://teddyrookbook.com/math/probability/Recent content in Probability on Teddy Rook Bookhttps://teddyrookbook.com/img/share.jpghttps://teddyrookbook.com/img/share.jpgHugoenhttps://teddyrookbook.com/math/probability/axioms/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/probability/axioms/Every probability rule follows from three: P(A) ≥ 0, P(S) = 1, and the probabilities of disjoint events add. Probability · Math Ladderhttps://teddyrookbook.com/math/probability/inclusion-exclusion-events/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/probability/inclusion-exclusion-events/P(A ∪ B ∪ C) adds the three probabilities, subtracts the three pairwise overlaps and adds back the triple overlap. Probability · Math Ladderhttps://teddyrookbook.com/math/probability/birthday-problem/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/probability/birthday-problem/Find "at least one match" through its complement, a product of shrinking fractions: in a group of 23, a shared birthday is slightly more likely than not. Probability · Math Ladderhttps://teddyrookbook.com/math/probability/conditional-vs-joint/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/probability/conditional-vs-joint/P(A ∩ B) is the chance that both happen; P(A ∣ B) is the chance of A among the outcomes where B happens. Read the question to see which one it asks for. Probability · Math Ladderhttps://teddyrookbook.com/math/probability/total-probability/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/probability/total-probability/Split into cases A₁, …, Aₙ that cover every outcome without overlapping: P(B) = Σ P(B ∣ Aᵢ) P(Aᵢ). Probability · Math Ladderhttps://teddyrookbook.com/math/probability/bayes-several/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/probability/bayes-several/P(Aⱼ ∣ B) = P(B ∣ Aⱼ) P(Aⱼ) / Σ P(B ∣ Aᵢ) P(Aᵢ), and each new piece of evidence updates the answer again. Probability · Math Ladderhttps://teddyrookbook.com/math/probability/independence-several/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/probability/independence-several/A, B and C are independent when every pair and the triple multiply; disjoint events with positive probabilities are never independent. Probability · Math Ladderhttps://teddyrookbook.com/math/probability/pmf-cdf/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/probability/pmf-cdf/A pmf lists P(X = x); the CDF F(x) = P(X ≤ x) climbs in steps, and P(a < X ≤ b) = F(b) − F(a). Probability · Math Ladderhttps://teddyrookbook.com/math/probability/expectation-of-function/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/probability/expectation-of-function/E[g(X)] = Σ g(x) P(X = x), and in general E[g(X)] ≠ g(E[X]). Probability · Math Ladderhttps://teddyrookbook.com/math/probability/variance-rules/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/probability/variance-rules/Var(X) = E[X²] − (E[X])², and Var(aX + b) = a² Var(X). Probability · Math Ladderhttps://teddyrookbook.com/math/probability/linearity-indicators/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/probability/linearity-indicators/E[X + Y] = E[X] + E[Y] always, so a count has mean equal to the sum of its indicators' probabilities: a shuffled deck has on average 1 card left in its starting place. Probability · Math Ladderhttps://teddyrookbook.com/math/probability/hypergeometric/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/probability/hypergeometric/Drawing n from N items, K of them successes: P(X = k) = C(K, k) C(N − K, n − k) / C(N, n). Probability · Math Ladderhttps://teddyrookbook.com/math/probability/negative-binomial/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/probability/negative-binomial/The number of trials needed for r successes has P(X = k) = C(k − 1, r − 1) pʳ (1 − p)^(k − r) and mean r/p. Probability · Math Ladderhttps://teddyrookbook.com/math/probability/poisson/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/probability/poisson/Counts of rare events in a fixed time or space: P(X = k) = e^(−λ) λᵏ / k!, with mean and variance both λ. Probability · Math Ladderhttps://teddyrookbook.com/math/probability/poisson-approximation/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/probability/poisson-approximation/When n is large and p is small, Binomial(n, p) is close to Poisson with λ = np. Probability · Math Ladderhttps://teddyrookbook.com/math/probability/choosing-distribution/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/probability/choosing-distribution/Match the story to binomial, geometric, negative binomial, hypergeometric or Poisson. Probability · Math Ladderhttps://teddyrookbook.com/math/probability/densities/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/probability/densities/A density f ≥ 0 has total area 1, and P(a ≤ X ≤ b) = ∫ₐᵇ f(x) dx. Probability · Math Ladderhttps://teddyrookbook.com/math/probability/continuous-cdf/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/probability/continuous-cdf/F(x) = ∫ f(t) dt from −∞ to x, f = F′, and the median m solves F(m) = 1/2. Probability · Math Ladderhttps://teddyrookbook.com/math/probability/continuous-expectation/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/probability/continuous-expectation/E[X] = ∫ x f(x) dx and E[g(X)] = ∫ g(x) f(x) dx, and the variance shortcut still holds. Probability · Math Ladderhttps://teddyrookbook.com/math/probability/exponential/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/probability/exponential/f(x) = λe^(−λx) for x ≥ 0 has mean 1/λ, and it is memoryless: P(X > s + t ∣ X > s) = P(X > t). Probability · Math Ladderhttps://teddyrookbook.com/math/probability/poisson-process/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/probability/poisson-process/For random arrivals at rate λ, the count in a window of length t is Poisson with mean λt, and the gaps between arrivals are exponential with mean 1/λ. Probability · Math Ladderhttps://teddyrookbook.com/math/probability/transform-density/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/probability/transform-density/Find the density of Y = g(X) through its CDF: work out F_Y(y) = P(g(X) ≤ y), then differentiate. Probability · Math Ladderhttps://teddyrookbook.com/math/probability/joint-pmf/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/probability/joint-pmf/A joint table gives P(X = x, Y = y); add across a row or down a column to get the marginal distributions. Probability · Math Ladderhttps://teddyrookbook.com/math/probability/conditional-distributions/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/probability/conditional-distributions/P(X = x ∣ Y = y) divides a row of the joint table by the row's total; X and Y are independent when every cell equals the product of its marginals. Probability · Math Ladderhttps://teddyrookbook.com/math/probability/geometric-probability/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/probability/geometric-probability/When a point is uniform in a region, probability is area over total area: two numbers chosen uniformly from [0, 1] differ by less than 1/4 with probability 7/16. Probability · Math Ladderhttps://teddyrookbook.com/math/probability/joint-densities/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/probability/joint-densities/P((X, Y) ∈ R) = ∬_R f(x, y) dA; set up the limits for regions such as {x < y}. Probability · Math Ladderhttps://teddyrookbook.com/math/probability/marginal-densities/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/probability/marginal-densities/f_X(x) = ∫ f(x, y) dy, and X and Y are independent when f(x, y) = f_X(x) f_Y(y) everywhere. Probability · Math Ladderhttps://teddyrookbook.com/math/probability/covariance-correlation/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/probability/covariance-correlation/Cov(X, Y) = E[XY] − E[X]E[Y], and the correlation ρ = Cov(X, Y)/(σ_X σ_Y) lies between −1 and 1. Probability · Math Ladderhttps://teddyrookbook.com/math/probability/variance-of-sums/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/probability/variance-of-sums/Var(X + Y) = Var(X) + Var(Y) + 2 Cov(X, Y), and the variance of a longer sum adds every pairwise covariance. Probability · Math Ladderhttps://teddyrookbook.com/math/probability/convolution/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/probability/convolution/The distribution of X + Y comes from adding over every split: P(X + Y = s) = Σ P(X = k) P(Y = s − k), with an integral for densities. Probability · Math Ladderhttps://teddyrookbook.com/math/probability/conditional-expectation/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/probability/conditional-expectation/E[X ∣ Y = y] averages X within one slice, and E[X] = E[E[X ∣ Y]] averages the slices. Probability · Math Ladderhttps://teddyrookbook.com/math/probability/min-max/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/probability/min-max/P(max ≤ x) = F(x)ⁿ and P(min > x) = (1 − F(x))ⁿ; the minimum of independent exponential variables is exponential with the rates added. Probability · Math Ladderhttps://teddyrookbook.com/math/probability/markov-chebyshev/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/probability/markov-chebyshev/For X ≥ 0, P(X ≥ a) ≤ E[X]/a; for any X with mean μ and standard deviation σ, P(|X − μ| ≥ kσ) ≤ 1/k². Probability · Math Ladderhttps://teddyrookbook.com/math/probability/law-large-numbers/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/probability/law-large-numbers/The average of n independent trials has variance σ²/n, so it settles near μ as n grows. Probability · Math Ladderhttps://teddyrookbook.com/math/probability/mgf/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/probability/mgf/M(t) = E[e^(tX)]; its derivatives at t = 0 give E[X], E[X²], …, and the MGF of a sum of independent variables is the product of their MGFs. Probability · Math Ladderhttps://teddyrookbook.com/math/probability/clt-sums/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/probability/clt-sums/A sum of n independent copies of X is close to normal, with mean nμ and standard deviation σ√n. Probability · Math Ladderhttps://teddyrookbook.com/math/probability/continuity-correction/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/probability/continuity-correction/Approximate Binomial(n, p) with a normal curve, widening each whole number to an interval of width 1: P(X ≤ 12) ≈ P(Y < 12.5). Probability · Math Ladderhttps://teddyrookbook.com/math/probability/markov-chains/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/probability/markov-chains/The next state depends only on the current one; a row of probabilities times the transition matrix P steps forward, so after n steps the distribution is π₀Pⁿ. Probability · Math Ladderhttps://teddyrookbook.com/math/probability/first-step-analysis/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/probability/first-step-analysis/Condition on the first step to get equations for the chance of absorption and the expected time to get there, as in the gambler's ruin. Probability · Math Ladderhttps://teddyrookbook.com/math/probability/stationary-distribution/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/probability/stationary-distribution/Solve πP = π together with π₁ + ⋯ + πₙ = 1 to find the long-run share of time spent in each state. Probability · Math Ladder