Mathematics, 26.02.2020 05:38 DWASS
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) 20 ≤ x ≤ 30, 20 ≤ y ≤ 30 0 otherwise (a) What is the value of K? (Enter your answer as a fraction.) K = 3/380000 Correct: Your answer is correct. (b) What is the probability that both tires are underfilled? (Round your answer to four decimal places.) .430426 Correct: Your answer is correct. (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.) .3593 Correct: Your answer is correct. (d) Determine the (marginal) distribution of air pressure in the right tire alone.
Answers: 1
![\int\limits_x {\int\limits_y {f\left( {x,y} \right)dydx} } = 1\\\int\limits_x {\left( {\int\limits_y {f\left( {x,y} \right)dy} } \right)dx} = 1\\K\int\limits_x {\left( {\int\limits_y {\left( {{x^2} + {y^2}} \right)dy} } \right)dx} = 1\\K\int\limits_x {\left( {\left[ {{x^2}y + \dfrac{{{y^3}}}{3}} \right]_{20}^{30}} \right)dx} = 1](/tpl/images/0524/6516/ecd4c.png)
![K\int\limits_x {\left( {10{x^2} + \dfrac{{19000}}{3}} \right)dx} = 1\\K\left[ {10\dfrac{{{x^3}}}{3} + \dfrac{{19000}}{3}x} \right]_{20}^{30} = 1\\K\left( {10\dfrac{{19000}}{3} + \dfrac{{19000}}{3}\left( {10} \right)} \right) = 1\\\dfrac{{380000}}{3}K = 1\\K = \dfrac{3}{{380000}}](/tpl/images/0524/6516/7e678.png)
![\\ = K\left[ {\left\{ {\int\limits_{28}^{30} {\int\limits_{x - 2}^{30} {\left( {{x^2} + {y^2}} \right)dydx} } } \right\} + \left\{ {\int\limits_{20}^{22} {\int\limits_{20}^{x + 2} {\left( {{x^2} + {y^2}} \right)dydx} } } \right\} + \left\{ {\int\limits_{22}^{28} {\int\limits_{x - 2}^{x + 2} {\left( {{x^2} + {y^2}} \right)dydx} } } \right\}} \right]\\ = K\left[ {\left\{ {\dfrac{{14804}}{{15}}} \right\} + \left\{ {\dfrac{{8204}}{{15}}} \right\} + \left\{ {\dfrac{{46864}}{{15}}} \right\}} \right]](/tpl/images/0524/6516/9b2df.png)
![= \dfrac{3}{{38000}}\left[ {\left\{ {\dfrac{{14804}}{{15}}} \right\} + \left\{ {\dfrac{{8204}}{{15}}} \right\} + \left\{ {\dfrac{{46864}}{{15}}} \right\}} \right]\\ = \dfrac{{4267}}{{11875}}\\ = 0.3593](/tpl/images/0524/6516/98b48.png)
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Each front tire on a particular type of vehicle is supposed to be filled to a pressure of 27 psi. Su...
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