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Specify the random ingredients
Let X be standard normal. Independently draw S as +1 or −1 with equal probability and set Y=SX. Symmetry of the normal density means Y also has a standard normal marginal. Both marginal means are zero and variances one.
Check the cross moment
Cov(X,Y)=E[SX²]=E[S]E[X²]=0. Thus the marginal moments and zero correlation resemble those of an independent standard normal pair, but this construction does not make X and Y independent.
Inspect the actual sum law
| Sign S | Sum X+Y | Probability |
|---|---|---|
| −1 | 0 exactly | 50% |
| +1 | 2X | 50% |
Identify the incompatible atom
The sum has variance 2, but at least 50% probability is concentrated exactly at zero. A positive-dispersion normal law is continuous and has zero mass at a single point. The sum therefore is not normal, despite both marginals being normal and covariance zero.
Keep moments separate from the joint law
A pair variance calculation remains valid for this model. Applying normal portfolio-tail formulas would add a distributional assumption that the construction contradicts.