In class, we analyzed Buffon's needle experiment. We showed that if a large sheet of paper has parallel lines that are 1 inch apart, and we throw a needle of length 1/2 inch at it, the probabilit...
In class, we analyzed Buffon's needle experiment. We showed that if a large sheet of paper has parallel lines that are 1 inch apart, and we throw a needle of length 1/2 inch at it, the probability that the needle hits a line is l/r. We can estimate π by throwing many needles and seeing how many throws hit a line. Suppose we throw a needle n times, and each throw is independent. Let X be the number of throws that hit a line. Our estimate for π is then Y = n/x. Let us use variance to figure out how many times we need to throw the needle to get a reasonably accurate estimate for π (a) Calculate E [X (b) Calculate Var [1/Y]. (c) Assuming that we have already determined that 3 ss 4, one can show that iflY-찌21/106, then l1/Y-1/n12 1/( 16x 106), This follows from the fact that l 1/Y-1/n1-1 Use (1) and Chebyshev's inequality to derive an upper bound on p(IY 1/10) in terms of n. How many throws of the needle do we need to ensure that plY-저 1 /1ofs 1/ 100? ecsll di e later in the course.)
In class, we analyzed Buffon's needle experiment. We showed that if a large sheet of paper has parallel lines that are 1 inch apart, and we throw a needle of length 1/2 inch at it, the probability that the needle hits a line is l/r. We can estimate π by throwing many needles and seeing how many throws hit a line. Suppose we throw a needle n times, and each throw is independent. Let X be the number of throws that hit a line. Our estimate for π is then Y = n/x. Let us use variance to figure out how many times we need to throw the needle to get a reasonably accurate estimate for π (a) Calculate E [X (b) Calculate Var [1/Y]. (c) Assuming that we have already determined that 3 ss 4, one can show that iflY-찌21/106, then l1/Y-1/n12 1/( 16x 106), This follows from the fact that l 1/Y-1/n1-1 Use (1) and Chebyshev's inequality to derive an upper bound on p(IY 1/10) in terms of n. How many throws of the needle do we need to ensure that plY-저 1 /1ofs 1/ 100? ecsll di e later in the course.)