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Consider a point (X, Y) E R² with joint density for some C≥ 0: f(x, y) = Ce (x, y) = R². x²+y (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. (c) 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).

Consider a point (X, Y) E R² with joint density for some C≥ 0: f(x, y) = Ce (x, y) = R². x²+y (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. (c) 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).

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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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, O are independent. (c) 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).
Transcribed Image Text: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, O are independent. (c) 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).
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