Mathematical Methods for Physics and Engineering - Matematica.NET

PROBABILITYθlighthouseLbeamOcoastlineyFigure 30.8The illumination of a coastline by the beam from a lighthouse.◮A lighthouse is situated at a distance L from a straight coastline, opposite a point O, andsends out a narrow continuous beam of light simultaneously in opposite directions. The beamrotates with constant angular velocity. If the random variable Y is the distance along thecoastline, measured from O, of the spot that the light beam illuminates, find its probabilitydensity function.The situation is illustrated in figure 30.8. Since the light beam rotates at a constant angularvelocity, θ is distributed uniformly between −π/2 and π/2, and so f(θ) =1/π. Nowy = L tan θ, which possesses the single-valued inverse θ =tan −1 (y/L), provided that θ liesbetween −π/2 andπ/2. Since dy/dθ = L sec 2 θ = L(1 + tan 2 θ)=L[1 + (y/L) 2 ], from(30.58) we findg(y) = 1 dθπ ∣ dy ∣ = 1for −∞