This problem describes a very useful way to generate independent and identically distributed (i.i.d.) Normal random...
Fantastic news! We've Found the answer you've been seeking!
Question:
Transcribed Image Text:
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
Expert Answer:
Answer rating: 100% (QA)
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
Related Book For
Mathematical Statistics With Applications In R
ISBN: 9780124171138
2nd Edition
Authors: Chris P. Tsokos, K.M. Ramachandran
Posted Date:
Students also viewed these mathematics questions
-
Suppose in Example 5.19 that a car can overtake a slower moving car without any loss of speed. Suppose a car that enters the road at time s has a free travel time equal to to. Find the distribution...
-
Let X1, . . . ,Xn be a set of independent and identically distributed continuous random variables having distribution function F, and let X(i), i = 1, . . . , n denote their ordered values. If X,...
-
Let X and Y be independent and identically distributed random variables with pdf Find the distribution of (XY)/2. 1x20 otherwise 0,
-
An object is placed on the principal axis of a concave mirror of focal length 10 cm at a distance of 80 cm from the pole. Find the position and the nature of the image.
-
Examine the cash flow measures in requirement 2 of the review problem in this chapter. Discuss the meaning of these ratios.
-
Rose Company has a relevant range of production between 10,000 and 25,000 units. The following cost data represents average cost per unit for 15,000 units of production. Using the cost data from Rose...
-
For maximum blade efficiency of single stage impulse turbine (a) \(ho\left(=\frac{u}{v_{a 1}} ight)=\frac{\cos ^{2} \alpha_{1}}{2}\) (b) \(ho=\cos \alpha_{1}\) (c) \(ho=\frac{\cos \alpha_{1}}{2}\)...
-
Soft Sopt is a manufacturer of futon mattresses. Soft Spots mattresses are priced at $60, but competition forces the company to offer significant discounts and rebates. As a result, the average price...
-
20 MPa 40 MPa Problem 1 For the given state of stress, determine (a) the orientation of the planes of maximum in-plane shearing stress, (b) the maximum in-plane shearing stress and the corresponding...
-
Should Grand Central Publishing continue to bid for Dewey, wait for the scheduled auction with several interested publishers, or pull out of the race? Why? 2. What is the role of authors, agents, and...
-
3. Given: F(s) = 3+6s+ 4s-2s s (s+3) Find the Inverse Laplace transform of F(s) using the Laplace Table. Show the entire solution.
-
From the market economics point of view, most of the commercial bioprocesses are operated in batch systems. Explain.
-
what journal entry is required when you pay your vendor within the discount period? Assume the perpetual inventory method.
-
Provide the main feature of the following operating systems? Multi-tasking or multi-programming batch operating system Time Sharing operating system Real-Time operating system
-
isk associated with marketing risk, credit risk, liquidity, and can also be a hostile takeover due to too much profit on hand is referred to as: a. Intangible risk b. Operating risk c. Financial risk...
-
What are some common causes of discrepancies between the bank statement and the company's records?
-
How do vaccines cause autism?
-
The tractor is used to lift the 150-kg load B with the 24-mlong rope, boom, and pulley system. If the tractor travels to the right at a constant speed of 4 m/s, determine the tension in the rope when...
-
Let X1, . . ., Xn be a random sample from a distribution with pdf Find the moment estimators for . 1+xx f(x, a) 1
-
Show that two events A and B are independent if and only if P (A B) = P (A) P (B) when at least one of P (A) or P (B) is not zero.
-
The following data give the weights (in pounds) for a random sample of 20 NFL players. Test the hypothesis that the median weight of NFL players is 250 pounds against the alternative that it is...
-
What factors is a judge likely to consider when determining best interest?
-
What is the best interest standard?
-
What criticisms are commonly made about the best interest standard?
Study smarter with the SolutionInn App