Both positive when the normals are correlated

Question

Let X and Y be standard normal random variables that are jointly normal with correlation ho = 	frac12 (each is standard, but they are no longer independent). What is P(X 0 	ext{ and } Y 0)?

Accepted Answer

For a jointly normal pair with correlation ho, the probability that both are positive is given by the orthant identity P(X 0,\; Y 0) = \frac14 + \frac{\arcsin ho}{2\pi}. Plug in ho = 	frac12, using \arcsin	frac12 = 	frac{\pi}{6}: P = \frac14 + \frac{\pi/6}{2\pi} = \frac14 + \frac{1}{12} = \frac{3}{12} + \frac{1}{12} = \frac{4}{12} = \frac13. The positive correlation makes "both positive" more likely than the independent value of 	frac14, which makes sense: when the variables tend to move together, they tend to share a sign. The identity behaves sensibly at the extremes: ho = 0 giv…

Both positive when the normals are correlated

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