?Consider X and Y two random variables of probability densities \(p_{1}(x)\) and

Chapter 5, Problem 119

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Consider X and Y two random variables of probability densities \(p_{1}(x)\) and \(p_{2}(x)\), respectively. The random variables X and Y are said to be independent if their joint density function is given by \(p(x, y)=p_{1}(x) p_{2}(y)\). At a drive-thru restaurant, customers spend, on average, 3 minutes placing their orders and an additional 5 minutes paying for and picking up their meals. Assume that placing the order and paying for/picking up the meal are two independent events X and Y. If the waiting times are modeled by the exponential probability densities

\(p_{1}(x)=\left\{\begin{array}{ll}\frac{1}{3} e^{-x / 3} & x \geq 0 \\0 & \text { otherwise }\end{array}\right.\)      and      \(p_{2}(y)=\left\{\begin{array}{ll}\frac{1}{5} e^{-y / 5} & y \geq 0 \\0 & \text { otherwise }\end{array}\right.\)

respectively, the probability that a customer will spend less than 6 minutes in the drive-thru line is given by \(P[X+Y \leq 6]=\iint_{D} p(x, y) d x d y\), where \(D=\{(x, y)\} \mid x \geq 0, y \geq 0, x+y \leq 6\}\) . Find \(P[X+Y \leq 6]\) and interpret the result.

Text Transcription:

P[X + Y leq 6] = iint_D p(x, y) dx dy

D = {(x, y)} mid x geq 0, y geq 0, x+y leq 6

P[X + Y leq 6]