$X$ and $Y$ have a bivariate normal distribution with $\rho$ as covariance. $X$ and $Y$ are standard normal variables.

I showed that $X$ and $Z= \dfrac{Y-\rho X}{\sqrt{1-\rho^2}}$ are independent standard normal variables.

Using this I need to show that

$$P(X >0,Y>0) = \frac14 + \frac{1}{2\pi} \cdot\arcsin(\rho).$$

  • $\begingroup$ As a hint for the first question, notice that $Z$ is the conditional of $Y$ on $X$. $\endgroup$ – Learner Dec 10 '12 at 11:03
  • $\begingroup$ it is easy to solve part. As X and Y are standard normal variables. But I am stuck at part two. $\endgroup$ – user669083 Dec 10 '12 at 11:10
  • $\begingroup$ If it is easy for part 1, why are you asking the question? $\endgroup$ – Learner Dec 10 '12 at 11:10
  • $\begingroup$ Because the second part is to use first part to solve second. $\endgroup$ – user669083 Dec 10 '12 at 11:12
  • $\begingroup$ @user669083 Learner is right. If you want an answer for part II, then you shouldnt say: "I had to show..." You need to say "using the fact that .....", I need to show : "part II" $\endgroup$ – Seyhmus Güngören Dec 10 '12 at 12:23

Let $\left( V, W \right)$ be i.i.d standard normal, let $U$ be uniform on $\left[ - \pi, \pi \right)$ independent of $R = \sqrt{V^2 + W^2}$ and let $\varphi = \arcsin \rho$. Then the following vectors have the same distributions (think about the Box-Muller transformation) \begin{eqnarray*} \left( X, Y \right) & \overset{d}{=} & \sqrt{2} \left( V \cos \varphi + W \sin \varphi, W \right)\\ \left( V, W \right) & \overset{d}{=} & R \left( \cos U, \sin U \right) \end{eqnarray*} Implying \begin{eqnarray*} P \left[ X > 0, Y > 0 \right] & = & P \left[ \cos U \cos \varphi + \sin U \sin \varphi > 0, \sin U > 0 \right]\\ & = & P \left[ U \in \left( \varphi - \frac{\pi}{2}, \varphi + \frac{\pi}{2} \right) \cap \left( 0, \pi \right) \right]\\ & = & \frac{\varphi}{2 \pi} + \frac{1}{4} \end{eqnarray*}

Here is an alternative proof:

Let $\phi$ be the density of the standard normal distribution \begin{eqnarray*} P \left[ X > 0, Y > 0 \right] & = & P \left[ X < 0, Y < 0 \right]\\ & = & \int_{- \infty}^0 \phi \left( z \right) \int_{- \infty}^0 \frac{1}{\sqrt{1 - \rho^2}} \phi \left( \frac{x - \rho z}{\sqrt{1 - \rho^2}} \right) \mathrm{d} x \mathrm{d} z\\ & = & \int_{- \infty}^0 \phi \left( z \right) \int_{- \infty}^{- \frac{\rho z}{\sqrt{1 - \rho^2}}} \phi \left( x \right) \mathrm{d} x \mathrm{d} z \end{eqnarray*} Let's call the above integral $h \left( \rho \right)$, then after some simplifications $$\frac{\partial h \left( \rho \right)}{\partial \rho} = \frac{1}{2 \pi \sqrt{1 - \rho^2}} $$ By integrating back (or considering the problem as a first-order ordinary differential equation), $h \left( \rho \right) = \frac{1}{2 \pi} \arcsin \rho + K$ where $K$ is some constant. By the special case of independence, $h \left( 0 \right) = \frac{1}{4}$, you get the final solution $$ P \left[ X > 0, Y > 0 \right] = \frac{1}{2 \pi} \arcsin \rho + \frac{1}{4}$$

  • $\begingroup$ is there a simpler solution ? $\endgroup$ – user669083 Dec 10 '12 at 17:58
  • $\begingroup$ @user669083 If you do not understand the above proof, you should point to the part where you have problems. Anyway, I am going to add an alternative proof (I am not sure it is simpler). $\endgroup$ – Learner Dec 11 '12 at 1:08
  • $\begingroup$ Let's call the above integral h(ρ), then after some simplifications. What is your simplification? Thanks in advance. $\endgroup$ – user52613 Dec 11 '12 at 9:38
  • $\begingroup$ @Fafsads Please check en.wikipedia.org/wiki/Leibniz_integral_rule for how to differentiate with respect to $\rho$. After that, all is needed is some simple manipulations of one-dimensional normal density integrals. Let me know after that if you are still having trouble. $\endgroup$ – Learner Dec 11 '12 at 10:34
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    $\begingroup$ I solved it by writing the equation as $(PX>0,Z> {\frac{-\rho X}{\sqrt{(1-\rho^2)}}})$. After that write in terms of integral, convert into polar co-ordinates and voila I got the answer $\endgroup$ – user669083 Dec 16 '12 at 9:52

Using the OP's hint: The event $\{X>0,Y>0\}$ is the same as the event $\{X>0,Z>\frac{-\rho}{\sqrt{1-\rho^2}}X\}$, where now $X$ and $Z$ are independent standard normal variables. Writing $a:=\frac{-\rho}{\sqrt{1-\rho^2}}$ for brevity, the desired probability is expressible as a double integral involving the joint density of $(X,Z)$: $$ P(X>0,Y>0)=P(X>0,Z>aX)=\int_{x=0}^\infty\int_{z=ax}^\infty\frac1{\sqrt{2\pi}}e^{-x^2/2}\frac1{\sqrt{2\pi}}e^{-z^2/2}\,dz\,dx. $$ Switching to polar coordinates, this equals $$ \begin{align} \int_{\theta=\arctan(a)}^{\pi/2}\int_{r=0}^\infty\frac1{2\pi}e^{-r^2/2}\,rdrd\theta&=\int_{\theta=\arctan(a)}^{\pi/2}\frac1{2\pi}\,d\theta\\ &=\frac1{2\pi}\left(\frac\pi2-\arctan a\right)\\ &=\frac14+\frac1{2\pi}\arctan\frac\rho{\sqrt{1-\rho^2}}; \end{align} $$ for the last equality we substitute $a:=\frac{-\rho}{\sqrt{1-\rho^2}}$ and use the fact that the arctan function is odd. To finish off, remember that if $\theta\in[-\pi/2,\pi/2]$ then $$\theta=\arctan\frac{\rho}{\sqrt{1-\rho^2}}\ \Longleftrightarrow\ \tan\theta=\frac{\rho}{\sqrt{1-\rho^2}}\ \Longleftrightarrow\ \sin\theta=\rho\ \Longleftrightarrow\ \theta=\arcsin \rho.$$


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