CMTH600- Assignment 1 - Due Wednesday, March 2 (in class) Page 1 of 2 1. In the \"hit-or-miss\" method, the value of is estimated as follows. A point is sampled uniformly in the square [1, 1] [1, 1], i.e. sample i.i.d. Cartesian coordinates X, Y Unif(1, 1). A point (X, Y ) is accepted if it lies inside the circle inscribed in the square, i.e. X 2 + Y 2 < 1. The experiment is repeated N times, and the number of accepted points, NH , is computed. The ratio NH converges to a limiting value, whose expression involves . N (a) Construct a random estimator of based on the experiment described above. (b) Construct a sample estimate of with N trials. Express it in terms of the ratio NH . N (c) Construct a 99%-condence interval for . Find the number of trials, N , required to obtain the condence interval of length less than 103 . 2. Let the point (X, Y ) be distributed uniformly in the circle with radius . The joint PDF 1 is fX,Y (x, y) = 2 1x2 +y2 <2 . by proceeding to the polar coordinates, nd joint pdf of radius r and angle show that are independent their marginal pdfs. propose an algorithm for sampling a point (r, ) uniformly distributed within circle. 3. let u unif(0, 1). m be positive integer. x : =M y {m } random variables. here is oor function, {x} fractional part function. 4. u1 , u2 un 1) (a) find cdfs then pdfs min(u1 max(u1 ). (b) 5. following generator ni =aNi1 mod m, with 216 + 3, 231 (1) sequence ui =Ni >