An ecologist wishes to select a point inside a circu- lar sampling region according to a uniform distribu- tion (in practice this could be done by rst selecting a direction and then a distance from the center in that direction). Let X = the x coordinate of the point0 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 den- sity D. Because f (x, y) is constant on D, comput- ing 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/ 12 replacing R/2.
d. What is the marginal pdf of X ? Of Y ? Are X and Y independent?