the distribution of a product and a quotient (ratio)
Not every combination of random quantities is a sum. Area is length times width; speed is distance divided by time; a ratio of two noisy readings is a quotient. When the variables that go in are random, the product XY and the quotient X/Y are random too, and their distributions can look surprisingly different from the inputs.
The clean way to find them is the multivariate transformation. To get the product, introduce a helper variable, say set W = XY and keep V = X, work out the joint density of (V, W) with the Jacobian, then integrate out the helper V to leave the density of W. The quotient works the same way: set R = X/Y, keep a helper, find the joint density, integrate out the helper. For independent variables this leads to tidy integrals: the density of the ratio is f_R(r) = integral over all y of f_X(r y) times f_Y(y) times |y| dy, the |y| coming straight from the Jacobian.
Ratios deserve respect because they misbehave. Dividing by a variable that can be near zero produces heavy tails and can destroy the mean entirely. The textbook shock: the ratio of two independent standard normals is a standard Cauchy distribution, which has NO mean and NO variance at all — its average never settles down no matter how much data you collect. So a ratio of perfectly well-behaved inputs can have no expectation. Treat sample ratios with care.
Let Z_1 and Z_2 be independent standard normals. Their ratio R = Z_1 / Z_2 has the standard Cauchy density f_R(r) = 1 / (pi (1 + r^2)). Although each Z is the tamest possible variable, the ratio has heavy tails and no mean: the chance of a huge ratio (when Z_2 lands near zero) decays too slowly for E[R] to exist.
A ratio of two well-behaved normals is Cauchy — heavy-tailed, with no mean at all.
A ratio can have no expectation even when both inputs do, because dividing by values near zero creates heavy tails. Never assume E[X/Y] equals E[X]/E[Y] — that is false in general.