Step 02 · Densities & Matrices

Gaussian Conditionals and Marginals

The bivariate normal density has a rare gift: every slice of it and every shadow of it is again a normal density. That self-similarity is what makes the Gaussian family the workhorse of the compendium.

Marginal shadow

Collapse the joint surface onto one axis by summing, or integrating, over the other. The result is the marginal density of the first quantity. For the bivariate normal the marginal is normal in its own right, centred at the first quantity's own mean and spread according to its own variance. Crucially, the marginal ignores the second quantity completely. Whatever association the two share leaves no mark on the shadow, because the shadow has already summed the second quantity away.

Conditional slice

Now hold one quantity fixed and slice the surface across the other. The slice, rescaled to unit area, is the conditional density. Here the normal family shows its most useful behaviour. The conditional density of the first quantity, given that the second equals some value, is again a normal density. Its mean shifts linearly with the fixed value, and its variance does not depend on the fixed value at all. In symbols, the conditional mean is the first mean plus a slope times the gap between the fixed value and the second mean, and the slope is the covariance divided by the variance of the fixed quantity.

Reading the rules in words

The conditional mean moves along a straight line as the fixed value changes.
The slope is the covariance divided by the variance of the quantity held fixed.
The conditional variance shrinks whenever the two quantities are associated.
The conditional variance does not change as the fixed value changes.

The third rule repays attention. Observing a companion quantity reduces the spread of the one you care about, and the reduction is larger when the association is stronger. In the extreme, if the two quantities moved in perfect step, the conditional variance would fall to zero and the slice would become a single point. When the association is absent, the conditional variance equals the marginal variance and the slice matches the shadow.

Two by two, then many

The same rules extend to many quantities at once. Partition a long list into the ones you observe and the ones you wish to guess. The conditional mean of the second part becomes a linear rule in the first part, and the conditional covariance becomes a smaller matrix obtained by subtracting a product of blocks. Every formula in the two-variable case survives as a block formula in the many-variable case, which is the deep reason the Gaussian family scales so gracefully.

Symmetry of the slice

Because the conditional density is normal, its slice is symmetric about the conditional mean. A reader can therefore sketch a conditional picture knowing only three numbers: the centre, the slope, and the conditional spread. That economy is why the normal family is used as a first model of dependence even when the data are not perfectly normal; the rules are clear, and departures are easy to describe against them.

An angled lecture board showing two axes and a family of parallel slice lines — Parallel slices across a joint surface: each slice is normal, and their centres climb a straight line.

Where this leads

Conditional and marginal views prepare the whole of the time-series material that follows. A stationary series, as the next step explains, is essentially a long sequence of quantities whose joint density is fixed by a simple rule, and the conditional instruments built here are exactly the instruments used to read it. A reader who can slice and collapse a two-variable picture has already learned the grammar of the later pages.