Add independent variables and change multiple coordinates

Try

Two independent Uniform(0,2) variables are added. Their sum lies between zero and four, but is not uniform: central sums have more contributing pairs. Sketch the square support and a diagonal strip x+y≈s to predict the shape of the sum density.

Predict

At s=1, the possible x values satisfying 0<x<2 and 0<s−x<2 form an interval of length one. At s=2 the length is two. Predict how the overlap length changes after s passes two. The moving support, not just the density height, creates the triangular law.

Experiment

The product joint density is 1/4. Therefore f_S(s)=s/4 for 0<s<2 and (4−s)/4 for 2<s<4, zero elsewhere. Each triangular half has area 1/2, verifying normalization.

Sum s	Allowed x interval	Density
1	(0,1)	1/4
2	(0,2)	1/2
3	(1,2)	1/4

The endpoints and the peak can be assigned any consistent density value without changing probabilities. Compare the sum’s mean, 2, with linearity of expectation as an independent check.

Understand

For independent continuous X,Y, the convolution formula is f_(X+Y)(s)=∫f_X(x)f_Y(s−x)dx. Without independence, use ∫f_(X,Y)(x,s−x)dx instead. Discrete convolution sums over pairs with the requested total. Independence justifies a product of marginals; zero covariance alone does not.

The formula can be derived by the coordinate change S=X+Y,T=X. Its inverse is X=T,Y=S−T, with absolute inverse Jacobian one. Transform the support as well as the density, then integrate out T. In the uniform example, the overlap bounds are max(0,s−2)<t<min(2,s), producing separate pieces.

For a general one-to-one smooth transformation of a density, multiply the original density at the inverse map by the absolute determinant of the inverse derivative. Global coverage and support must be checked. Several inverse branches contribute separate terms; determinant sign records orientation, not negative probability.

A useful example takes independent exponential rate-one X,Y and defines S=X+Y and U=X/(X+Y). The inverse is X=SU,Y=S(1−U), with S>0 and 0<U<1. Its absolute Jacobian is S. The joint density becomes S e^(−S), factorizing into a normalized Gamma(shape 2, rate 1) density for S and a uniform density for U. This proves independence of the transformed coordinates, not merely zero covariance.

The event X+Y=0 has probability zero in this exponential model, so the ratio’s undefined origin can be assigned arbitrarily without changing the law. Such null-set exceptions should be stated rather than hidden. In a model with an atom at zero, the same ratio would need separate treatment.

Convolution can describe sums even when no elementary antiderivative exists. A valid integral setup with correct support is a meaningful result; numerical approximation should be labeled and checked rather than replaced with an unsupported named family.

Apply

Derive the practice unit-square convolution from overlap bounds and verify total mass. For the sum-and-ratio transformation, compute the derivative matrix of the inverse explicitly and check its determinant. Then verify each factor integrates to one before announcing independence.

Continue

Next use moment-generating functions to identify certain sum laws. Their usefulness depends on existence around zero; finite moments alone are not enough.