statistics

b} Consider a two-dimension plane in which we marl: the lines 3.: = n for n E E . We now randomly "drop a needle\" [i.e. draw a line segment] of length 1 on the plane: its centre is given by two random co-ordinates {1L Y] , and the angle is given {in radians) by a random variable Ell . In this question, we will be concerned with the probability that the needle intersects one of the lines 3.! = n. For this purpose, we dene the random variable 3 as the distance from the needle's centre to the nearest line beneath it {i.e. Z = 1" - LY] , where [Y] is the greatest integer not greater than Y ]. We assume: - Z is uniformly distributed on [I], 1] . - o is uniformly distributed on [II], n] . I Z and B are independent and jointly continuous. i) |Give the density functions of Z and E. ii] Give the joint density function of E and E) [hint use the fact that E and B are independent]. lJroblen-i Sheet 3 Mathematics for AI By geometric reasoning, it can be shown that an intersection occurs it and only if: . 1 , 1 _ {2,3} E [0,1] x [H.112] is such that z 5 Esme? or 12 5 Esin iii) By using the joint distribution function of E and t3 , show that: 2 Pthe needle intersects a line} = . :1: Suppose now that a statistician is able to perform this experiment 1: times without any bias. Each drop of the needle is described by a random variable X,- wbich is 1 if the needle intersects a line and it otherwise. For any n , we assume the random variables X1, .. .,Jt'.',, are independent and identically distributed and that the variance of the population is o2 c oo