Question

Problem 5 . This question considers uniform random points on the unit disc x2+92 〈 1 (a) A point (X, Y) is uniformly chosen in the unit disc. Find the CDF and PDF of its distance from the origin R X2 +Y2 (b) Compute the expected distance from the origin. (c) Determine the marginal PDF of X and Y (d) Are X and Y independent? (Justify your claims) e) One way to generate uniform random points on this disc is to first generate uniform random points on the square-1. 1Ίκ [-1, 1] by selecting their coordinates independently, and ignoring points that lie outside the unit disc. To visualize this, generate 10000 uniform random points on the square and and create a scatter-plot of the points inside the disc (f) Another way to represent points in the plane is via polar coordinates (R cos Θ, R sin Θ) with R E 10, 1| and Θ E 10. 2πί. We might try naively to generate uniform random points in the circle by first generating a random radius R uniformly in [0, 1], and then by generating random ângle Θ uniformly in 10.2 . Generate 10000 such random pairs (R, Θ) and create a scatter-plot of the resulting points in the plane Does this look uniformly random? Compare the two plots.

Answer 1