# Report — F5-conf-03: sums of normally distributed random variables

# Question

Proposition A: if X and Y are each normal, then X+Y is normal (a constant counts as degenerate normal). Proposition B: if X and Y are independent and each normal, then X+Y is normal. Decide each; give a proof sketch or a concrete counterexample.

# Data and exclusions

Inputs: data/propositions.csv (2 rows) and data/design.csv (2 rows: seed 20261029, n 100000) — n_input_rows, random_seed. No data issues (data_issues is empty). No exclusions: all simulated rows are analysed (n_analysed = 100000; out/exclusions.csv). Hashes in out/state/data_audit.md.

# Method

Verdicts are by mathematical argument (out/argument.md). B: characteristic-function product + uniqueness. A: explicit counterexample X ~ N(0,1), S independent Rademacher, Y = S·X (counterexample_values.construction). The trivial Y = −X is rejected because its constant sum counts as normal under A's convention. The illustration simulates n draws with numpy.random.default_rng(seed) (out/analysis.py, out/counterexample.csv), and reports sample means/SDs (ddof=1) and the KS distance between the empirical law of x+y and the normal with its sample mean and SD. As a baseline, the same KS statistic is computed for the sum of two independent N(0,1) samples (baseline_indep_ks_sum_vs_normal).

# Results

Proposition A is false (statement_true = false); counterexample provided (counterexample_provided = true).
Proposition B is true (statement_b_true = true).
Simulated marginals: mean_x ≈ −0.003, sd_x ≈ 0.998, mean_y ≈ −0.0002, sd_y ≈ 0.998 (counterexample_values.mean_x, .sd_x, .mean_y, .sd_y); KS distances to fitted normals ≈ 0.0021 and 0.0020 (diagnostics.ks_x_vs_normal, diagnostics.ks_y_vs_normal), below the asymptotic 5% critical value ≈ 0.0043 (diagnostics.ks_critical_5pct).
Sum: mean ≈ −0.004, sd ≈ 1.406 (counterexample_values.mean_sum, .sd_sum; theory 0 and √2); KS distance to fitted normal ≈ 0.251 (counterexample_values.ks_sum_vs_normal), driven by the atom: 50.05% of sums are exactly 0 (diagnostics.frac_sum_exactly_zero; theory 1/2).
Baseline: independent normal sum has KS ≈ 0.0020 (baseline_indep_ks_sum_vs_normal), sd ≈ 1.412 (baseline_indep_sd_sum).
Sample correlation of x and y ≈ −0.008 (diagnostics.corr_xy): uncorrelated yet dependent, so uncorrelated normal marginals are not enough.

# Uncertainty

The verdicts are deterministic mathematical results and carry no statistical uncertainty. Simulated statistics carry Monte Carlo error of order 1/√n ≈ 0.003; the sum's KS distance (≈ 0.25) is about 60 times the 5% critical value, and theory gives a lower bound of 1/4 for any fitted normal because of the jump of size 1/2 at 0. (KS p-values with estimated parameters would be anti-conservative—Lilliefors issue—so they are not used as evidence.)

# Limitations and open issues

What the illustration establishes: on this seeded sample, x and y look N(0,1) while x+y has a 50% atom at 0 and is far from any normal—consistent with the proof.
What it does not establish: a finite sample cannot prove that the marginals are exactly normal or that the sum is non-normal; those facts rest on the argument in out/argument.md. The B baseline simulation likewise does not prove B.
The KS-based marginal check is a single test; it would not detect tiny departures, which is irrelevant here since marginal normality of Y is proved by symmetry.

spb/wp29_uqo