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 γ-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 given below 0 otherwise (a) What is the probability that the selected point is within of the center of the circular region? [Hint: Draw a picture of 5 the region of positive density D. Because f(x, y) is constant on D, computing a probability reduces to computing an area 25 (b) What is the probability that both X and Y differ from o by at most ? 7 318 (c) What is the probability that both X and Y differ from 0 by at most? 842
