- #1
dionysian
- 53
- 1
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 [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.
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.