Physics data analysis — the application of probability, statistics, and inference to experimental physics data. Distinguished from general machine-learning by two constraints: (i) data-generation mechanism is typically a known (or…
data-analysis-physics
Likelihood ratio: Λ = L(θ_1|x)/L(θ_0|x)
The likelihood ratio Λ = L_1/L_0 is the Neyman-Pearson 1933 most-powerful test statistic for simple-vs-simple hypothesis testing: given…
χ²(k) moments: E[χ²]=k, Var[χ²]=2k
The chi-square distribution χ²(k) with k degrees of freedom is the distribution of Σ_i^k Z_i² where Z_i ~ iid N(0,1). Probability density:…
Bootstrap distinct-fraction: 1 - (1-1/n)^n → 1 - 1/e
The bootstrap (Efron 1979) estimates sampling-distribution statistics by resampling n times with replacement from the original n data…
LR null: Λ(L_1=L_0) = 1
Sympy-exact witness of the likelihood-ratio null-state identity. Setup: Λ = L_1/L_0 as a two-symbol rational function. Identity: at L_1 =…
χ²(k=1) anchor: mean=1, variance=2
Sympy-exact witness of the k=1 chi-square distribution moments. Setup: E[χ²_k] = k, Var[χ²_k] = 2·k as one-symbol linear functions of k. …
Bootstrap distinct-limit: lim_{n→∞} [1-(1-1/n)^n] = 1-e^{-1}
Sympy-exact witness of the bootstrap distinct-fraction infinite-n limit. Setup: distinct-fraction = 1 - (1 - 1/n)^n as a function of n. …