Question

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 fx.y) =K(x+ y2) 20 s x s 30, 20 s y s 30 0 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.) at most 2 psi? (Round your answer to four decimal places.) (c) What is the probability that the difference in air pressure between the two tires (d) Determine the (marginal) distribution of air pressure in the right tire alone. for 20 xs 30 (e) Are X and Y independent rv's? Yes, f(x,y) fx(x) fy(y), so X and are independent. Yes, f(x,y) fx(x) - fy(y), so X and are independent No, f(x,y) fx(x) - fy(y), so X and Y are not independent No, f(x,y) fx(x) - fy(y), so X and Y are not independent.

Answer 1

a) for this to be valid:
c30 20

c30 20
f(x,y) dx dy =1

therefore k=3/380000

b)P(X<=27 , Y<=27) =0.4304

c) P(|X-Y|<=2) =0.3593

d) marginal distribution of X =f(x) =k*(19000/3+10x²)=(1/2+3x²/38000)