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 adirection 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. Ifthe circle is centered at (0, 0) and has radius R, then the joint pdf of X and Y is given below.1F(x, y) =x2 + y2 s R2otherwise(a) What is the probability that the selected point is within 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(c) What is the probability that both X and Y differ from 0 by at most(d) What is the marginal pdf of X?fxx) =What is the marginal pdf of Y?fkv) =Are X and Y independent?

*Answer: (b) Let denote the region described in the problem statement. Begin by explicitly writing…

Answered: iform distribution (in practice this…

A First Course in Probability (10th Edition)

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iform distribution (in practice this could be done by first selecting a te of the point selected and Y = the y coordinate of the point selected. If below. cular region? [Hint: Draw a picture of the region of positive density D. en area.]