1. Consider the Buffon needle problem. Let D be the distance between adjacent parallel lines on a flat plane. Further,

let ? be the acute angle between the parallel lines and the needle whose length is L

Let us assume that ? and Y are independent variables. We wish to compute the probability that if the needle is droppedon the plane, it crosses one of the lines. (You may make the same symmetry assumptions we made in class.)(a) Find the parameter a.(b) Find fY (y).(c) Find the joint PDF f(y, ?).(d) Write down the condition between y, L, ?, and D that corresponds to the needle crossing a line.(e) Make a sketch in the ?-y plane that shows the curve you found in (d).(f) Set up and evaluate the necessary integral to find the probability that the needle crosses a line.(g) Compare your result to two liming cases: (i) where ? is a uniform random variable, and (ii) where ? = ?/2, iethe needle is perpendicular to the parallel lines. Does your answer in (f) make sense?

I. Consider the Buffon needle problem. Let D be the distance between adjacent parallel lines on a flat plane. Further, let @ be the acute angle between the parallel lines and the needle whose length is &