1

Let $(X,Y)$ be bivariate normal with mean 0 and correlation coefficient $\rho$. Let $\beta$ be such that $ \cos \beta = \rho$, $(0\leq \beta \leq \pi)$, and show that $P(XY<0)=\frac{\beta}{\pi}$.

We are told to use the following details from the previous exercise: $Z=\frac{X}{Y}$ and $z=\rho \frac{\sigma_{X}}{\sigma_{Y}}$ ($\sigma_{X}=\sigma_{Y}=1$), and the density function $f_{Z}(z)=\frac{1}{\pi}\frac{\sqrt{1-\rho^{2}}}{(1-\rho^{2})+(z-\rho)^{2}}$.

First off, is there a typo in the question, and should it be $\sin(\beta) = \rho$ instead of $\cos(\beta)=\rho$?

Also, I keep getting an extra $+\frac{1}{2}$ at the end, so I need some major help with this problem. I thank you in advance!

  • 1
    Since $Y \neq 0$ almost surely, $P(XY <0) = P(\frac{X}{Y} <0)=P(Z<0)$. And you know the density function of $Z$... – Petite Etincelle Nov 12 '13 at 17:51
  • Yeah, I know that. I actually got the right answer literally 15 minutes ago. Getting cosine depends on using an inverse trig identity, $\arccos x = \frac{\pi}{2}-\arcsin x$, at the end. –  Nov 12 '13 at 19:30
  • https://math.stackexchange.com/questions/255368/px0-y0-for-a-bivariate-normal-distribution-with-correlation-rho. – StubbornAtom Mar 28 '19 at 11:40

0 Answers0