Two normals and a tilted quadrant

Question

Let X and Y be independent standard normal random variables. What is P(X 0 	ext{ and } X + Y 0)?

Accepted Answer

The joint density of (X,Y) is proportional to e^{-(x^2+y^2)/2}, rotationally invariant, so the probability of any wedge (cone) with apex at the origin is just its angle divided by 2\pi. Each condition is a half-plane through the origin: - \{X 0\}: points whose angle lies in (-90^\circ, 90^\circ). - \{X + Y 0\}: the half-plane on the positive side of the line y = -x, i.e. angles in (-45^\circ, 135^\circ). Their intersection is the wedge of angles in (-45^\circ, 90^\circ), an opening of 135^\circ: P = \frac{135^\circ}{360^\circ} = \frac{3}{8}. Cross-check via the arcsine formula. X and Z = \fra…

Two normals and a tilted quadrant

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