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

## Question
Decide whether (A) "X, Y each normal ⇒ X + Y normal" and (B) "X, Y independent and each normal ⇒ X + Y normal" are true; give a proof for true statements and a concrete, simulated counterexample for false ones.

## Data and exclusions
`data/propositions.csv` (2 rows: A, B) and `data/design.csv` (2 rows: seed = 20261029, n = 100000). Both read in full; no exclusions, no data issues found.

## Method
The verdicts rest on the mathematical argument in `out/argument.md`. Counterexample to A: X ~ N(0,1), W an independent random sign (±1, prob. ½ each), Y = W·X. Y ~ N(0,1) by symmetry of the normal law, while X + Y = (1+W)X equals 0 with probability ½ and has variance 2, so it is neither a non-degenerate normal (which has no atoms) nor a constant. B is proved with characteristic functions: φ_X φ_Y is a normal characteristic function.
`out/analysis.py` draws n = 100000 pairs with `numpy.random.default_rng(20261029)` (first X, then W), writes `out/counterexample.csv`, and computes sample moments (ddof = 1) and the KS distance of x + y from the normal with the same sample mean and sd.

## Results
- **A: false** (`statement_true = false`); a counterexample is provided.
- **B: true** (`statement_b_true = true`).
- Simulated sample (n = 100000): 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 ≈ 1.4142).
- KS distance of x + y vs. fitted normal: **0.2508**. For comparison, the same statistic for the marginals is 0.0021 (x) and 0.0020 (y), against an approximate 5% critical value of 1.358/√n ≈ 0.0043 (this is the known-parameter value; the Lilliefors value for estimated parameters is smaller, ≈0.0028, and the marginal statistics are below that too). 50.05% of the simulated sums are exactly 0 (theory: 50%).

## Uncertainty
The verdicts are deductive, so they carry no sampling uncertainty. The simulated numbers are subject to Monte Carlo error of order 1/√n ≈ 0.003 for the means and the KS statistics. The KS distance of the sum (0.25) is about 60 times the 5% critical value, so the departure from normality cannot be put down to sampling noise. The marginal KS values are consistent with normality.

## Limitations and open issues
The numerical illustration is consistent with the proof: the marginals look normal, and the sum has a point mass at 0 and is far from normal. It does **not** prove anything. A finite sample can neither establish that the marginals are exactly normal nor, on its own, rule out normality in general. Both conclusions come from the argument in `argument.md`. The KS statistic uses estimated parameters, so the standard KS p-values do not apply exactly, which is why no p-value is reported. Many other counterexamples exist (e.g. Y = X·sign(c − |X|)); this one was chosen because its non-normality (an atom of mass ½) is very easy to see.