Bivariate Normal probability question

Question

I have this homework question

Suppose $(X,Y)\sim BN(u_x=0,u_y=0,w_x^2=1,w_y^2=1,p=-0.6)$.

Find:

a) $c$ such that $8X+10Y$ and $cX+5Y$ are independent

b) $P(X<0,Y>0)$

My thoughts are

(a) do we use $E(XY)-E(X)E(Y)=0$?

(b) is that $P(x<0)=1/2=p(y>0)$?

Answer 1

(a) since $(8X+10Y, cX+5Y)$ will be bivariate normal, this means that $8X+10Y$ and $cX+5Y$ will be independent if and only if $Cov(8X+10Y,cX+5Y)=0$ thus from properties of covariance we have to solve $$Cov(8X+10Y,cX+5Y)=(8c)Var(X)+(8)(5)Cov(X,Y)+(10c)Cov(X,Y)+5(10)Var(Y)=0$$

(b) Though yes $P(X<0)=P(Y>0)=\frac{1}{2}$, this does not really tells us anything about what $P(X<0,Y>0)$ since $\rho\neq 0$ we have that $X$ and $Y$ are not independent. This one you will most likely have to do using the pdf explicitly, at line (71) http://mathworld.wolfram.com/BivariateNormalDistribution.html it shows what this comes out to explicitly. on page 556 line (4) http://alcatel-lucent.com/bstj/vol38-1959/articles/bstj38-2-553.pdf it gives a hint on how to do this