Solution Found!
Solution: In Exercise 5.16, Y1 and Y2 denoted the
Chapter 5, Problem 109E(choose chapter or problem)
In Exercise \(Y_{1}\) and \(Y_{2}\) denoted the proportions of time that employees 1 and II actually spent working on their assigned tasks during a workday. The joint density of \(Y_{1}\) and \(Y_{2}\) is given by
\(f\left(y_{1}, y_{2}\right)=\left\{\begin{array}{ll}
y_{1}+y_{2} & 0 \leq y_{1} \leq 1,0 \leq y_{2} \leq 1, \\
0, & \text { elsewhere. }
\end{array}\right.
\)
In Exercise , we derived the mean of the productivity measure \(30 Y_{1}+25 Y_{2}\). Find the variance of this measure of productivity. Give an interval in which you think the total productivity measures of the two employees should lie for at least
of the days in question.
Equation Transcription:
Text Transcription:
Y_1
Y_2
Y_1
Y_2
f(y_1,y_2)={_0, elsewhere. ^y_1+y_2, 0</=y_1</=1, 0</=y_2</=1,
30Y_1+25Y_2
Questions & Answers
QUESTION:
In Exercise \(Y_{1}\) and \(Y_{2}\) denoted the proportions of time that employees 1 and II actually spent working on their assigned tasks during a workday. The joint density of \(Y_{1}\) and \(Y_{2}\) is given by
\(f\left(y_{1}, y_{2}\right)=\left\{\begin{array}{ll}
y_{1}+y_{2} & 0 \leq y_{1} \leq 1,0 \leq y_{2} \leq 1, \\
0, & \text { elsewhere. }
\end{array}\right.
\)
In Exercise , we derived the mean of the productivity measure \(30 Y_{1}+25 Y_{2}\). Find the variance of this measure of productivity. Give an interval in which you think the total productivity measures of the two employees should lie for at least
of the days in question.
Equation Transcription:
Text Transcription:
Y_1
Y_2
Y_1
Y_2
f(y_1,y_2)={_0, elsewhere. ^y_1+y_2, 0</=y_1</=1, 0</=y_2</=1,
30Y_1+25Y_2
ANSWER:
Answer:
Step 1 of 1:
In Exercise
and
denoted the proportions of time that employees
actually spent working on their assigned tasks during a workday.
The joint probability density function of and
is given by,
The Employee has a higher productivity rating than employee
and a measure of the total productivity of the pair of employees is
Find the variance of this measure of productivity.
Give an interval in which you think the total productivity measures of the two employees should lie for at least 75% of the days in question.
We need to find
Let and
jointly continuous random variables with the joint (or bivariate) probability function
Then the marginal density functions of
and
respectively, are given by
Hence the marginal density functions for is,
The marginal density functions for is,
Let and
jointly continuous random variables with the joint (or bivariate) probability function
Then the expected value of
and
respectively, are given by