Each front tire on a particular type of vehicle is supposed to be filled to a pressure of 27 psi. Suppose the actual air pressure in each tire is a random variable, X for the right tire and Y for the left tire, with joint pdf f(x,y) = {5( K(x² + y²) 20≤x≤ 30, 20 ≤ y ≤ 30 otherwise (a) What is the value of K? (Enter your answer as a fraction.) K= (b) What is the probability that both tires are underfilled? (Round your answer to four decimal places.) (c) What is the probability that the difference in air pressure between the two tires is at most 2 psi? (Round your answer to four decimal places.) (d) Determine the (marginal) distribution of air pressure in the right tire alone. for 20≤x≤ 30 (e) Are X and Y independent rv's? OYes, f(x, y) = fx(z) - fy(y), so X and Y are independent. OYes, f(x, y) + fx(x) fy(y), so X and Y are independent. ONO, f(x, y) = fx(x) fy(y), so X and Y are not independent. ONO, f(x, y) + fx(x) fy(y), so X and Y are not independent.
Each front tire on a particular type of vehicle is supposed to be filled to a pressure of 27 psi. Suppose the actual air pressure in each tire is a random variable, X for the right tire and Y for the left tire, with joint pdf f(x,y) = {5( K(x² + y²) 20≤x≤ 30, 20 ≤ y ≤ 30 otherwise (a) What is the value of K? (Enter your answer as a fraction.) K= (b) What is the probability that both tires are underfilled? (Round your answer to four decimal places.) (c) What is the probability that the difference in air pressure between the two tires is at most 2 psi? (Round your answer to four decimal places.) (d) Determine the (marginal) distribution of air pressure in the right tire alone. for 20≤x≤ 30 (e) Are X and Y independent rv's? OYes, f(x, y) = fx(z) - fy(y), so X and Y are independent. OYes, f(x, y) + fx(x) fy(y), so X and Y are independent. ONO, f(x, y) = fx(x) fy(y), so X and Y are not independent. ONO, f(x, y) + fx(x) fy(y), so X and Y are not independent.
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
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Transcribed Image Text:Each front tire on a particular type of vehicle is supposed to be filled to a pressure of 27 psi. Suppose the actual air pressure in each tire is a random variable, X for the right tire and Y for the left tire, with joint pdf
f(x, y) =
K(x² + y²)
{K
0
20≤x≤ 30, 20 ≤ y ≤ 30
otherwise
(a) What is the value of K? (Enter your answer as a fraction.)
K=
(b) What is the probability that both tires are underfilled? (Round your answer to four decimal places.)
(c) What is the probability that the difference in air pressure between the two tires is at most 2 psi? (Round your answer to four decimal places.)
(d) Determine the (marginal) distribution of air pressure in the right tire alone.
for 20 ≤ x ≤ 30
(e) Are X and Y independent rv's?
OYes, f(x, y) = fx(x) · fy(y), so X and Y are independent.
OYes, f(x, y) + fx(x) · fy(y), so X and Y are independent.
ONO, f(x, y) = fx(x) · fy(y), so X and Y are not independent.
ONO, f(x, y) + fx(x) fy(y), so X and Y are not independent.
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