Solved: Each rear tire on an experimental airplane is

Chapter 4, Problem 3.44

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Each rear tire on an experimental airplane is supposed to be filled to a pressure of 40 pounds per square inch (psi). Let X denote the actual air pressure for the right tire and Y denote the actual air pressure for the left tire. Suppose that X and Y are random variables with the joint density function

\(f(x, y)=\left\{\begin{array}{ll} k\left(x^{2}+y^{2}\right), & 30 \leq x<50,30 \leq y<50, \\ 0, & \text { elsewhere, } \end{array}\right.\)

(a) Find k.

(b) Find \(P(30\ \leq\ X\ \leq\ 40\) and \(40\ \leq\ Y\ <\ 50\)).

(c) Find the probability that both tires are underfilled.

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