1# Report — F5-conf-03: sums of normally distributed random variables23## Question4Proposition A: if X and Y are each normally distributed, X + Y is normal (constants count as degenerate normals).5Proposition B: if X and Y are independent and each normal, X + Y is normal.6Decide each by mathematical argument; give a concrete, simulated counterexample for any false proposition.78## Data and exclusions9Inputs: `data/propositions.csv` (2 rows) and `data/design.csv` (2 rows: seed 20261029, n 100000) — `n_input_rows`, `random_seed`. No missing values, duplicates or anomalies (`data_issues` = []); no exclusions (`out/exclusions.csv`). The simulated sample has `n_analysed` = 100000 rows.1011## Method12- Verdicts are established by proof (full argument in `out/argument.md`), not by simulation.13- B: characteristic-function argument; independence factorises φ_{X+Y} = φ_X φ_Y = exp(i(μ₁+μ₂)t − (σ₁²+σ₂²)t²/2).14- A: counterexample X ~ N(0,1), W independent Rademacher sign, Y = W·X (`counterexample_values.construction`). Y ~ N(0,1) by symmetry; X+Y = (1+W)X has an atom of mass 1/2 at 0 and is not constant, so it is neither a non-degenerate nor a degenerate normal. (Y = −X would not work, since the constant sum counts as normal under A.)15- Illustration: `numpy.random.default_rng(seed)` with design seed and n; draw x then w; save (x, y) to `out/counterexample.csv`. Report means and sds (ddof = 1) and the KS distance of x+y to the normal with the sample mean and sd. As a baseline (the B setting), the same KS rule is applied to x + z with z an independent N(0,1) draw (`supplementary.baseline_ks_independent_sum_vs_normal`).1617## Results18- Proposition A is **false** (`statement_true` = false); a concrete counterexample is provided (`counterexample_provided` = true).19- Proposition B is **true** (`statement_b_true` = true).20- Simulated sample (`counterexample_values`): mean_x = −0.0033, sd_x = 0.9982; mean_y = −0.0002, sd_y = 0.9982; mean_sum = −0.0036, sd_sum = 1.4059 (theory √2 ≈ 1.414).21- KS of x+y vs fitted normal: `counterexample_values.ks_sum_vs_normal` = 0.2508, versus 0.0021 and 0.0020 for the marginals (`supplementary.ks_x_vs_normal`, `supplementary.ks_y_vs_normal`) and 0.0018 for the independent-sum baseline (`supplementary.baseline_ks_independent_sum_vs_normal`).22- Share of sums exactly equal to 0: `supplementary.share_sum_exactly_zero` = 0.50047 (theory 0.5). Sample correlation of x and y: `supplementary.corr_x_y` = −0.0082 (theory 0): the two normals are uncorrelated but dependent.23- `feasibility` = "complete".2425## Uncertainty26The verdicts are mathematical and carry no statistical uncertainty. The sample statistics have sampling error of order 1/√n ≈ 0.003 (e.g. se of mean_x ≈ 0.003, of sd_sum ≈ 0.005 roughly); all observed values are within that range of their theoretical values. The KS statistics use parameters estimated from the sample (Lilliefors setting), so standard KS p-values do not apply; we report distances only. The population KS distance of the sum to any normal is ≥ 0.25 (atom of mass 1/2), matching the observed 0.2508.2728## Limitations and open issues29- What the illustration establishes: for this seed and n, the simulated x and y are numerically indistinguishable from N(0,1) by KS distance, while x+y is grossly non-normal (half its mass at exactly 0; KS ≈ 0.25, about 100× the marginal values). It is a reproducible concrete instance consistent with the proof.30- What it does not establish: a finite sample cannot prove that a marginal is exactly normal or that a law is non-normal; the truth values of A and B rest solely on the arguments in `out/argument.md`. The simulation also does not test proposition B beyond one illustrative baseline.31- `supplementary` keys go beyond the required schema; they are diagnostics only.32- No open issues.33