Isye 2027

# If w g x and we wish to nd the pdf of w at a

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: le problem. displacement of the needle is sin(Θ). Let U denote the vertical distance from the lower endpoint of the needle to the ﬁrst grid line above that endpoint. It is reasonable to assume that U is uniformly distributed from 0 to 1, and that U is independent of Θ. The needle intersects the grid if and only 1 if U ≤ sin(Θ). The joint density of (Θ, U ), is equal to π over the rectangular region [0, π ] × [0, 1]. Integrating that density over the region {(θ, u) : u ≤ sin(θ)} shown in Figure 4.18 yields u 1 0 " 0 ! Figure 4.18: Region of integration for Buﬀon needle problem. 142 CHAPTER 4. JOINTLY DISTRIBUTED RANDOM VARIABLES P {needle intersects grid} = P {U ≤ sin(Θ)} π sin θ 1 dudθ π 0 0 π dθ cos(θ) sin(θ) = π π 0 = = π = 0 2 ≈ 0.6366. π The answer can also be written as E [sin(Θ)]. A way to interpret this is that, given Θ, the probability the needle intersects a line is sin(Θ), and averaging over all Θ (in a continuous-type version of the law of total probability) gives the overall answer. Example 4.6.2 Consider the follow...
View Full Document

## This note was uploaded on 02/09/2014 for the course ISYE 2027 taught by Professor Zahrn during the Spring '08 term at Georgia Tech.

Ask a homework question - tutors are online