Solution
The Nearest Star
Show the problem again
Points are scattered over the plane according to a homogeneous Poisson process with an average of 1 point per unit area. What is the expected distance from the origin to the nearest point? Give a reduced fraction.
Worked solution
The answer is 1/2. The nearest point is farther than r exactly when the disk of radius r around the origin contains no points. That disk has area πr², so the number of points in it is Poisson with mean πr², and P(D > r) = e^(−πr²). Then E[D] = ∫₀^∞ e^(−πr²) dr = (1/2)·√(π/π) = 1/2, using ∫₀^∞ e^(−ar²) dr = (1/2)√(π/a). Rescaling, at density λ the expected nearest distance is 1/(2√λ). The naive guess that the nearest neighbour is about one "typical spacing" 1/√λ away overestimates by a factor of 2.
Source: Classic nearest-neighbour calculation for a planar Poisson process, in wide circulation; no traceable original source. Statement written for AxiomIQ.