This problem describes a very useful way to generate independent and identically distributed (i.i.d.) Normal random...

  

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This problem describes a very useful way to generate independent and identically distributed (i.i.d.) Normal random variables. Let (X, Y) be i.i.d. N (0, 1) random variables. Let R be the radius vector corresponding to the point (X, Y) and let be the angle R subtends with the positive direction of the x-axis. Thus (R, 0) represents the vector (X, Y) in polar coordinate and we have (see Figure 1) X = R Cos 0, Y = R Sin 0. a. What are the possible values of R and O? (5 points) b. Find the joint density of (R, 0). (5 points) C. Find the marginal densities of R and Ⓒ. (8 points) d. Find the density of R². (5 points) e. Are R² and independent? (2 points) [Hint: Note that you are given the inverse transformation rather than the transformation, i.e. you are given X = g₁¹(R,O) and Y = g2¹ (R,) as opposed to usual formulation (R, 0) = g (X, Y).] y 0 T r cos Figure 1 r sin 0 This problem describes a very useful way to generate independent and identically distributed (i.i.d.) Normal random variables. Let (X, Y) be i.i.d. N (0, 1) random variables. Let R be the radius vector corresponding to the point (X, Y) and let be the angle R subtends with the positive direction of the x-axis. Thus (R, 0) represents the vector (X, Y) in polar coordinate and we have (see Figure 1) X = R Cos 0, Y = R Sin 0. a. What are the possible values of R and O? (5 points) b. Find the joint density of (R, 0). (5 points) C. Find the marginal densities of R and Ⓒ. (8 points) d. Find the density of R². (5 points) e. Are R² and independent? (2 points) [Hint: Note that you are given the inverse transformation rather than the transformation, i.e. you are given X = g₁¹(R,O) and Y = g2¹ (R,) as opposed to usual formulation (R, 0) = g (X, Y).] y 0 T r cos Figure 1 r sin 0

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a The possible values of R and depend on the range of the random variables X and Y Since X and Y are independent and identically distributed standard normal random variables they can take any real val... View the full answer