Report — F5-conf-03: sums of normal random variables
Question
Is 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.
Data and exclusions
data/propositions.csv: 2 rows (A, B).data/design.csv: 2 rows (seed = 20261029, n = 100000).- No data issues were found and nothing was excluded.
Method
- The verdicts come from mathematical argument (see
out/argument.md). The simulation only illustrates them. - 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.
- B (true). Proof via characteristic functions: multiply them, then apply the uniqueness theorem.
- Simulation (
out/analysis.py):numpy.random.default_rng(20261029). It draws n = 100000 values of X, then n signs (in that order), and writesout/counterexample.csv(columns x, y).- Reported statistics: sample means and SDs (ddof = 1) of x, y and x + y.
- KS distance of x + y against N(sample mean, sample SD).
- 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).
Results
- Verdicts: A is false (
statement_true = false; counterexample provided). B is true (statement_b_true = true). - Simulated counterexample (n = 100000, seed 20261029):
| quantity | value | theoretical |
|---|---|---|
| mean_x / sd_x | −0.0033 / 0.9982 | 0 / 1 |
| mean_y / sd_y | −0.0002 / 0.9982 | 0 / 1 |
| mean_sum / sd_sum | −0.0036 / 1.4059 | 0 / √2 ≈ 1.4142 |
| KS(x+y vs fitted normal) | 0.2508 | > 0 (the law has an atom of 1/2) |
| KS(x vs N(0,1)), KS(y vs N(0,1)) | 0.0028, 0.0016 | 0 |
| fraction of x+y exactly 0 | 0.5005 | 0.5 |
| corr(x, y) | −0.0082 | 0 |
- The marginals fit N(0,1) closely: both KS distances are below the approximate 5% critical value 1.36/√n ≈ 0.0043.
- 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.
Uncertainty
- The verdicts are mathematical facts and carry no statistical uncertainty.
- Sample statistics have Monte Carlo error of about 1/√n ≈ 0.003 for means. Their deviations from the theoretical values are consistent with that.
- 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.
Limitations and open issues
- 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.
- 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. - 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.