Suppose that (X, Y ) is uniform on the disk of radius 1 as in Example E of Section 3.3.W...

Suppose that (X, Y ) is uniform on the disk of radius 1 as in Example E of Section 3.3.Without doing any calculations, argue that X and Y are not independent.

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A point is chosen randomly in a disk of radius 1. Since the area of the disk is π,

We can calculate the distribution of R, the distance of the point from the origin. R ≤ r if the point lies in a disk of radius r . Since this disk has area πr ²,

The density function of R is thus f_R(r ) = 2r, 0 ≤ r ≤ 1.

Let us now find the marginal density of the x coordinate of the random point:

Note that we chose the limits of integration carefully; outside these limits the joint density is zero. (Draw a picture of the region over which f (x, y) > 0 and indicate the preceding limits of integration.) By symmetry, the marginal density of Y is