A point in a triangle

Topic: Joint distributions · more questions on this topic
‹ A product density Signs and their sum ›

A point (X,Y) is drawn from the triangle with vertices (0,0), (1,0) and (1,1) by this rule: draw X uniformly from [0,1], then draw Y uniformly from [0,X].

  1. Give the conditional density of Y given X=x, and say where it is zero.

  2. Find the joint density of (X,Y).

  3. Find the marginal density of Y.

  1. Given X=x, Y is uniform on [0,x], so f_{Y\mid X}(y\mid x)=1/x for 0\le y\le x and 0 otherwise.

  2. By the product rule for densities,

f_{X,Y}(x,y)=f_X(x)\,f_{Y\mid X}(y\mid x)=\frac1x,\qquad 0\le y\le x\le1,

and zero elsewhere. The point is not uniform on the triangle, whose uniform density would be the constant 2: it is crowded towards the vertex at the origin, where the thin vertical slices are.

  1. Integrating out x,

f_Y(y)=\int_y^1\frac{\,\mathrm{d} x}x=-\log y,\qquad 0<y<1.