X and Y are jointly normal, each with mean 0 and variance 1, and with correlation 1/2. What is P(X > 0, Y > 0)? Give the answer as a fraction in lowest terms.

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$X$ and $Y$ are jointly normal, each with mean $0$ and variance $1$ , and with correlation $\frac{1}{2}$ . What is $P (X > 0, Y > 0)$ ? Give the answer as a fraction in lowest terms.

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Hint

Treat $X$ and $Y$ as projections of a round normal cloud onto directions whose angle has cosine equal to $ρ$ .

Solution

Take independent standard normals $A$ and $B$ . The point $(A, B)$ has a uniformly random direction. Let $X$ be its projection on one unit vector and $Y$ its projection on another, at angle $θ$ with $cos (θ) = \frac{1}{2}$ , so $θ = π /3$ .
Each condition keeps a half plane, and the angle between the half planes' edges leaves a wedge of angle $π - π /3 = 2 π /3$ .
The probability is the share of the full circle, $\frac{2 π /3}{2 π} = \frac{1}{3}$ .

Answer: $\frac{1}{3}$

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