Problem 4*. Consider a point (X, Y) € R² with joint density for some C ≥ 0: f(x, y) = Ce-√x² + y² (x, y) = R². (a) Find

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Problem 4 Consider A Point X Y R With Joint Density For Some C 0 F X Y Ce X Y X Y R A Find 1 (126.73 KiB) Viewed 12 times

Problem 4*. Consider a point (X, Y) € R² with joint density for some C ≥ 0: f(x, y) = Ce-√x² + y² (x, y) = R². (a) Find the constant C. (b) Let (R, O) be the polar coordinates of (X, Y). Find the density of R, O. Show that R, are independent. What is the probability that (X, Y) lands on the 3rd quadrant? (d) Find the expected value and variance of R. (e) Let (X', Y') be another point on R2 with density f, which is independent of (X, Y). Find the probability that there is at least one of the point is landed in the ball B₁(0) of radius 1 centered at origin (0,0).