STAT 350: Lecture 16

Joint Densities

Suppose and are independent standard normals. In class I said that their joint density was

Here I want to show you the meaning of joint density by computing the density of a random variable.

Let . By definition U has a distribution with 2 degrees of freedom. The cumulative distribution function of U is

For this is 0 so take . The event that is the same as the event that the point is in the circle centered at the origin and having radius , that is, if A is the circle of this radius then

By definition of density this is a double integral

You do this integral in polar co-ordinates. Letting and we see that

The Jacobian of the transformation is r so that becomes . Finally the region of integration is simply and so that

The density of U can be found by differentiating to get

which is the exponential density with mean 2. This means that the density is really an exponential density.