Solution
Inside the Unit Circle
Show the problem again
A dart lands at (X, Y), where X and Y are independent standard normal random variables. What is the probability that the dart lands within distance 1 of the origin? Give an exact closed form, or a decimal rounded to 4 places.
Worked solution
The answer is 1 − e^(−1/2) ≈ 0.3935. The joint density (1/2π) e^(−(x² + y²)/2) depends only on the radius, so in polar coordinates P(R ≤ 1) = ∫₀^{2π} ∫₀¹ (1/2π) e^(−r²/2) r dr dθ = ∫₀¹ r e^(−r²/2) dr = 1 − e^(−1/2). More generally R² = X² + Y² is exponential with mean 2 (a chi-square with two degrees of freedom), which is the trick behind the Box-Muller method for generating normal variables. Note how much smaller this is than the one-dimensional figure P(|X| ≤ 1) ≈ 0.683.
Source: Classic Gaussian exercise in wide circulation; no traceable original source. Statement written for AxiomIQ.