An ecologist wishes to select a point inside a circular sampling region according to a u...

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

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 replacing R/√2.

d. What is the marginal pdf of X? Of Y? Are X and Y independent?