Given a bivariate standard-normally distributed random variable $Y=[Y_1,Y_2]^T$ with density $\phi(Y_i,Y_2)$, the probability of $Y_1>0$ is simply $$P(Y_1 > 0)=1- \int_{-\infty}^{\infty} \int_{-\infty}^{0} \phi(y_i,y_2) dy_1dy_2=\frac{1}{2}.$$ I try to find the integral of $\phi(Y_i,Y_2)$ under the constraint $Y_1+Y_2 \ge 0$, that is $P(Y_1+Y_2 \ge 0)$. How should I define the integral and integrate the bivariate density?
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2$\begingroup$ Could you first consider whether or not you can directly derive the distribution of $Y_1+Y_2$? $\endgroup$– Xi'anCommented Feb 29, 2016 at 13:30
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$\begingroup$ Useful. Seems one can sum (e.g., en.wikipedia.org/wiki/…) and then apply the integral given above on the distribution of the sum? $\endgroup$– tomkaCommented Feb 29, 2016 at 13:59
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1$\begingroup$ Not the integral above since it then is a unidimensional integral... $\endgroup$– Xi'anCommented Feb 29, 2016 at 14:02
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$\begingroup$ $y_1$ can take on any value in $\mathbb{R}$ but $y_2$ can be at most $-y_1$. This should help you set up the integral. $\endgroup$– dsaxtonCommented Feb 29, 2016 at 19:43
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