Solution Found!
The total time from arrival to completion of service at a
Chapter 6, Problem 10E(choose chapter or problem)
The total time from arrival to completion of service at a fast-food outlet, \(Y_{1}\), and the time spent
waiting in line before arriving at the service window, \(Y_{2}\), were given in Exercise 5.15 with joint
density function
\(f\left(y_{1}, y_{2}\right)=\left\{\begin{array}{ll} e^{-y_{1}}, & 0 \leq y_{2} \leq y_{1}<\infty \\ 0, & \text { elsewhere. } \end{array}\right.\)
Another random variable of interest is \(\mathrm{U}=Y_{1}-Y_{2}\), the time spent at the service window. Find
a the probability density function for U.
b \(\mathrm{E}(\mathrm{U})\) and \(\mathrm{V}(\mathrm{U})\). Compare your answers with the results of Exercise 5.108.
Equation Transcription:
{
Text Transcription:
Y_1
Y_2
f(y_1,y_2)= {e^-y_1 0 \leq y_2 \leq y_1 \leq \infty 0, elsewhere
U=Y_1-Y_2
U
E(U)
V(U)
Questions & Answers
QUESTION:
The total time from arrival to completion of service at a fast-food outlet, \(Y_{1}\), and the time spent
waiting in line before arriving at the service window, \(Y_{2}\), were given in Exercise 5.15 with joint
density function
\(f\left(y_{1}, y_{2}\right)=\left\{\begin{array}{ll} e^{-y_{1}}, & 0 \leq y_{2} \leq y_{1}<\infty \\ 0, & \text { elsewhere. } \end{array}\right.\)
Another random variable of interest is \(\mathrm{U}=Y_{1}-Y_{2}\), the time spent at the service window. Find
a the probability density function for U.
b \(\mathrm{E}(\mathrm{U})\) and \(\mathrm{V}(\mathrm{U})\). Compare your answers with the results of Exercise 5.108.
Equation Transcription:
{
Text Transcription:
Y_1
Y_2
f(y_1,y_2)= {e^-y_1 0 \leq y_2 \leq y_1 \leq \infty 0, elsewhere
U=Y_1-Y_2
U
E(U)
V(U)
ANSWER:
Solution:
Step 1 of 2:
The total time from arrival to completion of service at a fast -food outlet is Y1, and the time spent waiting in line before arriving at the service window is Y2.
The joint density function of Y1 and Y2 is given by,
The time spend at the service window, U= Y1-Y2.
We have to find,
- The probability density function for U.
- E(U) and V(U).