An ecologist wishes to select a point inside a circular sampling region according to a uniform distribution (in practice this could be done by first selecting a direction and then a distance from the center in that direction). Let X the x coordinate of the point selected and Y the y coordinate of the point selected. If the circle is centered at (0, 0) and has radius R, then the joint pdf of X and Y is f(x, y) { 1 R 2 x2 y2 R2 0 otherwise a. What is the probability that the selected point is within R/2 of the center of the circular region? [Hint: Draw a picture of the region of positive density D. Because f(x, y) is constant on D, computing a probability reduces to computing an area.] b. What is the probability that both X and Y differ from 0 by at most R/2? c. Answer part (b) for R/2 replacing R/2. d. What is the marginal pdf of X? Of Y? Are X and Y independent? 1

Assignment # 3 STA 5205, 5126 and 4202 Date: Wednesday, October 26, 2015 This assignment is based on Chapter 5. You must show all necessary work to get full credit. Please submit your assignment in due time. Some selected questions will be graded. Pl keep 1” margin in all sides of the paper. You must define both null and alternative hypotheses for any test related question. You may use any computer software unless oherwise stated. This is your cover page. Only hard copy of the assignemnt will be accepted. First & Last Name:----------------------------------- Panther ID:------------------------------------------- Problem #1: Exercise 5.1, page 225. Problem # 2: An engineer suspects that the surface finish of a metal part is