# CDF of conditional distribution

Suppose that $X$ and $Y$ are iid normally distributed and $a$ is a scalar. What is $\Pr(Y+aX<0 | X>0)$?

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Sorry about that -- I clarified that they're jointly normal. –  ABC Jul 26 '12 at 3:58
Try this paper: etrij.etri.re.kr/Cyber/servlet/… –  Douglas Zare Jul 26 '12 at 5:44
Transform $(X,Y)\rightarrow (X,Z)$, where $Z=Y+aX$; you will obtain that $(X,Z)$ has a bivariate normal distribution with a certain mean and covariance matrix. Then use the definition ${\mathbb P}(Y+aX<0\vert X>0)=\dfrac{{\mathbb P}(Y+aX<0,X>0)}{{\mathbb P}(X>0)}=\dfrac{{\mathbb P}(Z<0,X>0)}{{\mathbb P}(X>0)}$. The numerator can be calculated using the CDF of a bivariate normal and the denominator using the CDF of the univariate normal. –  user10525 Jul 26 '12 at 9:37
If the means are zero, this requires (virtually) no calculation at all. Are you interested in the general case or, perhaps, this specific one? –  cardinal Jul 26 '12 at 13:12
@cardinal I guess the OP is interested in the general case because this expression appears in the paper mentioned in the previous question. I also guess he did not mean i.i.d.. –  user10525 Jul 26 '12 at 13:20