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If Y1 is the total time between a customer’s arrival in
Chapter 5, Problem 108E(choose chapter or problem)
If \(Y_{1}\) is the total time between a customer's arrival in the store and departure from the service window and if \(Y_{2}\) is the time spent in line before reaching the window, the joint density of these variables was given in Exercise 5.15 to be
\(f\left(y_{1}, y_{2}\right)=\left\{\begin{array}{ll} e^{-y_{1}}, & 0 \leq y_{2} \leq y_{1} \leq \infty, \\ 0, & \text { elsewhere. } \end{array}\right.\)
The random variable \(Y_{1}-Y_{2}\) represents the time spent at the service window. Find \(E\left(Y_{1}-Y_{2}\right)\) and \(V\left(Y_{1}-Y_{2}\right)\). Is it highly likely that a randomly selected customer would spend more than 4 minutes at the service window?
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QUESTION:
If \(Y_{1}\) is the total time between a customer's arrival in the store and departure from the service window and if \(Y_{2}\) is the time spent in line before reaching the window, the joint density of these variables was given in Exercise 5.15 to be
\(f\left(y_{1}, y_{2}\right)=\left\{\begin{array}{ll} e^{-y_{1}}, & 0 \leq y_{2} \leq y_{1} \leq \infty, \\ 0, & \text { elsewhere. } \end{array}\right.\)
The random variable \(Y_{1}-Y_{2}\) represents the time spent at the service window. Find \(E\left(Y_{1}-Y_{2}\right)\) and \(V\left(Y_{1}-Y_{2}\right)\). Is it highly likely that a randomly selected customer would spend more than 4 minutes at the service window?
ANSWER:Step 1 of 5
If \(Y_{1}\) is the total time between a customer’s arrival in the store and departure from the service window and if \(Y_{2}\) is the time spent in line before reaching the window.
The joint probability density function of \(Y_{1}\) and \(Y_{2}\) is given by,
\(f\left(y_{1}, y_{2}\right)=\left\{\begin{array}{ll} e^{-y_{1}}, & 0 \leq y_{2} \leq y_{1} \leq \infty, \\ 0, & \text { elsewhere. } \end{array}\right.\)
The random variable\(Y_{1}-Y_{2}\) represents the time spent at the service window.
We need to find the value of \(E\left(Y_{1}-Y_{2}\right)\) and \(V\left(Y_{1}-Y_{2}\right)\).
Is it highly likely that a randomly selected customer would spend more than 4 minutes at the service window?
The marginal density functions for \(Y_{1}\) is,
\(f_{1}\left(y_{1}\right)=\int_{-\infty}^{\infty} f\left(y_{1}, y_{2}\right) d y_{2}\)
\(f_1\left(y_1\right)=\int_0^{y_1}e^{-y_1}dy_2=y_1e^{-y_1},\ \quad y_1\ge0\)
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