Student T Distribution | t distribution and gamma distribution
Example1.2AMoreGeneralBuffonsNeedleDerivationConsiderthecaseshowninthefiguretotheleftwhenoneendoftheneedleisdroppedatO,adistancexfromtheleftline.Becauseallorientationsoftheneedleareequallyprobable,theotherendoftheneedlecanbeanywhereonthecircleofradiusLcenteredatO.Theprobabilityaneedleintersectseithertherightorleftlineisjustthesumofthearclengthss1ands2outsidethestripdividedbythecirclescircumference2πL.Thearclengths1isLtimestheangleAOB,whichcanbeexpressedas(1.4)s1(x)={2Lcos−1(x/L),0
Consider the case shown in the figure to the left when one end of the needle is dropped at O, a distance x from the left line. Because all orientations of the needle are equally probable, the other end of the needle can be anywhere on the circle of radius L centered at O. The probability a needle intersects either the right or left line is just the sum of the arc lengths s1 and s2 outside the strip divided by the circles circumference 2π L. The arc length s1 is L times the angle AOB, which can be expressed as
(1.4)s1(x)={2Lcos−1(x/L),0<x<L0,x≥L.
Similarly,
(1.5)s2(x)={2Lcos−1((D−x)/L),x>D−L0,0<x≤D−L.
>The end of the needle can land anywhere in the strip with equal probability, so the probability the needle end is dropped in dx about x is simply dx/D. The probability a needle with an end at x intersects one of the bounding lines is [s1(x) + S2(x)]/(2π L). Hence, the probability a dropped needle...