Two perpendicular half-planes

Question

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

Accepted Answer

The joint density of (X, Y) is proportional to e^{-(x^2+y^2)/2}, which is rotationally invariant, so the probability of any wedge with apex at the origin is its opening angle divided by 360^\circ. Each condition is a half-plane through the origin: - \{X + Y 0\}: the side of the line y = -x where x + y 0. - \{X - Y 0\}: the side of the line y = x where x y. The two boundary lines y = -x and y = x are perpendicular, so their two "positive" half-planes intersect in a right-angle wedge, an opening of 90^\circ: P = \frac{90^\circ}{360^\circ} = \frac14. Cross-check via independence. Let U = X + Y a…

Two perpendicular half-planes

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