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.(adsbygoogle = window.adsbygoogle || []).push({});

Let [tex]X[/tex] and [tex]Y[/tex] be defined by:

[tex]\begin{array}{l}

X = \cos \theta \\

Y = \sin \theta \\

\end{array}[/tex]

Where [tex]\theta [/tex] is a uniform random variable distributed over [tex](0,2\pi )[/tex]

A) Show that [tex]X[/tex] and [tex]Y[/tex] are uncorrelated

Attempt at solution:

Show [tex]{\mathop{\rm cov}} (x,y) = 0[/tex]

[tex]\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}[/tex]

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

In the solution to the problem its seems that they assume that

[tex]{f_{xy}}(x,y) = {f_\theta }(\Theta )[/tex]

Then the integeral they use becomes

[tex]E[xy] = \int\limits_0^{2\pi } {xy{f_\theta }(\Theta )d\theta } [/tex]

But how come it is valid to assume that

[tex]{f_{xy}}(x,y) = {f_\theta }(\Theta )[/tex]

Doesn’t the joint (and the marginal) pdf change because of the functions:

[tex]\begin{array}{l}

X = \cos \theta \\

Y = \sin \theta \\

\end{array}[/tex]

I f anyone knows what I am trying to ask please give me a little help to what is going on here.

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# Joint expectation of two functions of a random variable

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