Joint expectation of two functions of a random variable

[dionysian]

Ok I am not sure if I should put this question in the homework category of here but it’s a problem from schaums outline and I know the solution to it but I don’t understand the solution 100% so maybe someone can explain this to me.
Let X and Y be defined by:
\begin{array}{l}<br /> X = \cos \theta \\ <br /> Y = \sin \theta \\ <br /> \end{array}
Where \theta is a uniform random variable distributed over (0,2\pi )
A) Show that X and Y are uncorrelated
Attempt at solution:
Show {\mathop{\rm cov}} (x,y) = 0
\begin{array}{l}<br /> {\mathop{\rm cov}} (x,y) = E[xy] - E[x]E[y] \\ <br /> E[xy] = \int\limits_0^{2\pi } {\int\limits_0^{2\pi } {xy{f_{xy}}(x,y)dxdy} } \\ <br /> \end{array}
Now my question is how do we determine the joint pdf {{f_{xy}}(x,y)} if we only know the marginal pdfs of \theta?
In the solution to the problem its seems that they assume that
{f_{xy}}(x,y) = {f_\theta }(\Theta )
Then the integeral they use becomes
E[xy] = \int\limits_0^{2\pi } {xy{f_\theta }(\Theta )d\theta }
But how come it is valid to assume that
{f_{xy}}(x,y) = {f_\theta }(\Theta )
Doesn’t the joint (and the marginal) pdf change because of the functions:
\begin{array}{l}<br /> X = \cos \theta \\ <br /> Y = \sin \theta \\ <br /> \end{array}
I f anyone knows what I am trying to ask please give me a little help to what is going on here.

 

[bpet]

... Now my question is how do we determine the joint pdf {{f_{xy}}(x,y)} if we only know the marginal pdfs of \theta?

The joint pdf doesn't technically exist, because the random variables (X,Y) have all their mass on a 1-dimensional subset of 2d space, i.e. the circle (cos(theta),sin(theta)). The joint pdf could be written in terms of Dirac delta functions or the expectation could be written as a Stieltjes integral (using the joint cdf) but the theory gets messy and for this example it's much simpler to write the expectation as

E[XY] = E[cos(\theta)sin(\theta)]

which can be expressed as a single integral because theta has a pdf.