Mathematics, 06.04.2021 03:30 enazer
A fast-food restaurant operates both a drive through facility and a walk-in facility. On a randomly selected day, let X and Y, respectively, be the proportions of the time that the drive-through and walk-in facilities are in use, and suppose that the joint density function of these random variables is,
f (x, y) ={2/3(x+2y) 0 ≤ x ≤ 1 , 0 ≤ y ≤ 1
(a) Find the marginal density of X.
(b) Find the marginal density of Y .
(c) Find the probability that the drive-through facility is busy less than one-half of the time.
Answers: 3
![(b)\ h(y) = \frac{1}{3}[1 + 4y]](/tpl/images/1240/9520/aec95.png)
![h(y) = \frac{2}{3}[(\frac{1^2}{2} + 2y*1) - (\frac{0^2}{2} + 2y*0) ]](/tpl/images/1240/9520/a20d9.png)
![h(y) = \frac{2}{3}[(\frac{1}{2} + 2y)]](/tpl/images/1240/9520/94d6b.png)
![h(y) = \frac{1}{3}[1 + 4y]](/tpl/images/1240/9520/80bc8.png)
![P(x0.5) =\frac{2}{3}\int\limits^1_0 [(\frac{0.5^2}{2} + 2*0.5y) -(\frac{0^2}{2} + 2*0y)], dy](/tpl/images/1240/9520/85839.png)
![P(x0.5) =\frac{2}{3}[(0.125*1 + \frac{1^2}{2}) - (0.125*0 + \frac{0^2}{2})]](/tpl/images/1240/9520/96231.png)
![P(x0.5) =\frac{2}{3}[(0.125 + \frac{1}{2})]](/tpl/images/1240/9520/8eb93.png)
![P(x0.5) =\frac{2}{3}[(0.125 + 0.5]](/tpl/images/1240/9520/043d2.png)
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A fast-food restaurant operates both a drive through facility and a walk-in facility. On a randomly...
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