1# Report — F5-conf-03: sums of normal random variables23## Question4Is X + Y necessarily normal when (A) X and Y are each normal, or (B) X and Y are independent and each normal? For each false proposition, give a concrete, verifiable counterexample.56## Data and exclusions7- `data/propositions.csv`: 2 rows (A, B). `data/design.csv`: 2 rows (seed = 20261029, n = 100000).8- No data issues were found and nothing was excluded.910## Method11- The verdicts come from mathematical argument (see `out/argument.md`). The simulation only illustrates them.12- **A (false).** Construction: X ~ N(0,1), S an independent random sign (±1), Y = S·X. Both marginals are N(0,1). X + Y equals 0 with probability 1/2 and 2X otherwise. This large atom at 0 makes the departure from normality obvious at any reasonable sample size.13- **B (true).** Proof via characteristic functions: multiply them, then apply the uniqueness theorem.14- **Simulation** (`out/analysis.py`): `numpy.random.default_rng(20261029)`. It draws n = 100000 values of X, then n signs (in that order), and writes `out/counterexample.csv` (columns x, y).15 - Reported statistics: sample means and SDs (ddof = 1) of x, y and x + y.16 - KS distance of x + y against N(sample mean, sample SD).17 - Auxiliary checks: KS distance of x and of y against N(0,1), the fraction of exact zeros in x + y, and corr(x, y).1819## Results20- Verdicts: **A is false** (`statement_true = false`; counterexample provided). **B is true** (`statement_b_true = true`).21- Simulated counterexample (n = 100000, seed 20261029):2223| quantity | value | theoretical |24|---|---|---|25| mean_x / sd_x | −0.0033 / 0.9982 | 0 / 1 |26| mean_y / sd_y | −0.0002 / 0.9982 | 0 / 1 |27| mean_sum / sd_sum | −0.0036 / 1.4059 | 0 / √2 ≈ 1.4142 |28| KS(x+y vs fitted normal) | **0.2508** | > 0 (the law has an atom of 1/2) |29| KS(x vs N(0,1)), KS(y vs N(0,1)) | 0.0028, 0.0016 | 0 |30| fraction of x+y exactly 0 | 0.5005 | 0.5 |31| corr(x, y) | −0.0082 | 0 |3233- The marginals fit N(0,1) closely: both KS distances are below the approximate 5% critical value 1.36/√n ≈ 0.0043.34- The sum's KS distance of 0.25 is about 58 times that critical value, and half of the sum values are exactly 0. A normal law cannot produce this.3536## Uncertainty37- The verdicts are mathematical facts and carry no statistical uncertainty.38- Sample statistics have Monte Carlo error of about 1/√n ≈ 0.003 for means. Their deviations from the theoretical values are consistent with that.39- The KS critical value above is the textbook value for a fully specified null. Because the sum's normal was fitted from the sample, that test (Lilliefors setting) is conservative. This does not matter here: the observed distance exceeds it by more than an order of magnitude.4041## Limitations and open issues42- **What the simulation establishes.** For this specific construction, a finite sample has marginals that look normal and a sum that is plainly non-normal, with an atom at 0 of about 1/2. This matches the theory and checks that the construction is implemented correctly.43- **What it does not establish.** A finite sample cannot prove that the marginals are exactly normal, and it cannot prove proposition B. Both rest on the proofs in `out/argument.md`.44- The counterexample is one of many. Any (X, Y) with normal marginals that is not jointly normal can give a non-normal sum, though not every such pair does.45