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.