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.