Stats Solutions

[Solution Library] (a) Let \Theta be a continuous random variable uniformly distributed on [0,2 π] . Let X= cos \Theta and Y= sin \Theta . Show that, for this

Question: (a) Let \(\Theta\) be a continuous random variable uniformly distributed on \([0,2 \pi] .\) Let \(X=\cos \Theta\) and \(Y=\) \(\sin \Theta .\) Show that, for this \(X\) and \(Y\), \(X\) and \(Y\) are uncorrelated but not independent.

(b) Recall that two random variables \(X\) and \(Y\) are jointly Gaussian if they have the joint probability density function:

\[f_{X, Y(x, y)}=\frac{1}{2 \pi \sigma_{X} \sigma_{Y} \sqrt{1-\rho^{2}}} \exp \left(\frac{-1}{2\left(1-\rho^{2}\right)}\left\{\left(\frac{x-\mu_{X}}{\sigma_{X}}\right)^{2}-2 \rho \frac{\left(x-\mu_{X}\right)}{\sigma_{X}} \frac{\left(y-\mu_{Y}\right)}{\sigma_{Y}}+\left(\frac{y-\mu_{Y}}{\sigma_{Y}}\right)^{2}\right\}\right)\]

where \(\mu_{X}\) is the mean of \(X, \mu_{Y}\) is the mean of \(Y, \sigma_{X}\) is the standard deviation of \(X, \sigma_{Y}\) is the standard deviation of \(Y\), and \(\rho\) is called the "correlation coefficient". From this monster equation, it is possible to show that:

\[\begin{aligned} &f_{X}(x)=\frac{1}{\sqrt{2 \pi \sigma_{X}^{2}}} \exp \left(-\frac{1}{2}\left(\frac{x-\mu_{X}}{\sigma_{X}}\right)^{2}\right) \\ &f_{Y}(y)=\frac{1}{\sqrt{2 \pi \sigma_{Y}^{2}}} \exp \left(-\frac{1}{2}\left(\frac{y-\mu_{Y}}{\sigma_{Y}}\right)^{2}\right) \end{aligned}\]

If \(X\) and \(Y\) are uncorrelated, then \(\rho=0 .\) Show that \(\rho=0\) implies, for jointly Gaussian \(X\) and \(Y\), that \(X\) and \(Y\) are independent.

Price: $2.99
Solution: The downloadable solution consists of 2 pages
Deliverable: Word Document


∴ Other downloads you may be interested in ∴

log in to your account

Forgot password?
Don't have a membership account?
REGISTER

reset password

Back to
log in

sign up

Back to
log in