I was studying random processes and couldn't solve this? Could you help?

"Let x and y be zero-mean, jointly Gaussian random variables. Assuming that Var(x)= $\sigma_x^2$and Var(y)= $\sigma_y^2$, find a scalar a in terms of variance of x and and $r_{xy}$ such that x-ay and y are independent random variables?"

I don't understand this: if x and y are independent then $f_x(x)f_y(y)=f_{xy}(xy)$ but how can I place x-ay?

  • 1
    $\begingroup$ The variables x and are jointly Gaussian so they can be correlated and in fact they have correlation $r_xy$. So find the a that makes them independent. As a hint: In addition to finding a that makes the product of the marginal densities the joint density you can just pick a to make them uncorrelated. $\endgroup$ – Michael R. Chernick Apr 4 '17 at 23:46

You need to know (and use) two facts about jointly Gaussian random variables.

  1. If $X$ and $Y$ are jointly Gaussian random variables, then, for all choices of real numbers $\alpha, \beta, \gamma$,and $\delta$, so also are $\alpha X + \beta Y$ and $\gamma X + \delta Y$ jointly Gaussian random variables.
  2. Jointly Gaussian random variables are independent if and only if they are uncorrelated random variables, or, equivalently, if they have zero covariance.

So, what can you say about the joint distribution of $X-aY$ and $Y$ given that $X$ and $Y$ are jointly Gaussian random variables? Regardless of the joint distribution of $X$ and $Y$, do you know (or can you figure out) how to calculate $\operatorname{cov}(X-aY, Y)$, the covariance of $X-aY$ and $Y$, in terms of $\sigma_X^2, \sigma_Y^2$ and $\operatorname{cov}(X,Y)$? Is there a value of $a$ for which $\operatorname{cov}(X-aY, Y)$ has value $0$?


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