# Joint expectation of two functions of a random variable

1. Sep 7, 2010

### 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:
$$\begin{array}{l} X = \cos \theta \\ Y = \sin \theta \\ \end{array}$$
Where $$\theta$$ is a uniform random variable distributed over $$(0,2\pi )$$
A) Show that $$X$$ and $$Y$$ are uncorrelated
Attempt at solution:
Show $${\mathop{\rm cov}} (x,y) = 0$$
$$\begin{array}{l} {\mathop{\rm cov}} (x,y) = E[xy] - E[x]E[y] \\ E[xy] = \int\limits_0^{2\pi } {\int\limits_0^{2\pi } {xy{f_{xy}}(x,y)dxdy} } \\ \end{array}$$
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:
$$\begin{array}{l} X = \cos \theta \\ Y = \sin \theta \\ \end{array}$$
I f anyone knows what I am trying to ask please give me a little help to what is going on here.

2. Sep 7, 2010

### bpet

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.