P: Conditional densities require a new support

A joint density can be constant without the variables being independent. A triangular support changes the range available to one variable when the other is fixed. Draw the region, obtain the marginal density and normalise within the conditional support.

Worked example or practice scenario

Let f(x,y)=2 on 0<y<x<1. Then f_X(x)=2x. Given X=x, Y is uniform on (0,x), so E[Y|X=x]=x/2. The conditional support is not (0,1).

Try this next

Integrate the conditional density over (0,x) to check it equals one. Then compute E[Y]=E[X]/2=1/3. If your answer uses a constant conditional mean of 1/2, revisit the geometry.

Reading sources

• Official syllabus and resources

ActNet editorial guide · October 1, 2026 · Original illustrative examples.