# Random Variables Study Example

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?

• 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. – Michael R. Chernick Apr 4 '17 at 23:46

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.
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 $0$?