# As in the special case of z y linear transformations

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Unformatted text preview: ributions are limits of binomial distributions. A similar approach was used in Examples 4.5.1 and 4.5.2. 4.6. ADDITIONAL EXAMPLES USING JOINT DISTRIBUTIONS 4.6 141 Additional examples using joint distributions Example 4.6.1 Buﬀon’s needle problem Suppose a needle of unit length is thrown at random onto a large grid of lines with unit spacing. Find the probability the needle, after it comes to rest, intersects a grid line. Solution: An important aspect of solving this problem is how to model it, and that depends on how we think about it. Here is one possible way to think about it–but there are others. Imagine that the needle lands in a plane with horizontal and vertical directions, and that the grid lines are horizontal. When the needle lands, let Θ, with 0 ≤ Θ ≤ π, denote the angle between the line going through the needle and a grid line, measured counter-clockwise from the grid line, as shown in Figure 4.17. Assume that Θ is uniformly distributed over the interval [0, π ]. The vertical sin(! ) U ! Figure 4.17: Solving the Buﬀon need...
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## This note was uploaded on 02/09/2014 for the course ISYE 2027 taught by Professor Zahrn during the Spring '08 term at Georgia Institute of Technology.

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