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Density of standard normal given sum larger than zero

Question

Given two i.i.d. standard normal random variables $X, Y$. What is the density of $X$ given $X+Y>0$?

Answer

Find the probability of $X\le c$ given $X+Y>0$.

\[\begin{align}P(X\le c\vert X+Y>0)&=\frac{P(X\le c,Y>-X)}{P(X+Y>0)}\\&=2\int_{-\infty}^c\int_{-X}^{+\infty}YdYdX\\&=2\int_{-\infty}^c\Phi(X)dX\\&=2\Phi(X)\phi(X)\end{align}\]

Thus the density of $X$ given $X+Y>0$ is:

\[P(X\le c\vert X+Y>0)=2\Phi(X)\phi(X)\]

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