Solution Found!
Answer: In Exercise 5.18, Y1 and Y2 denoted the lengths of
Chapter 5, Problem 81E(choose chapter or problem)
In Exercise 5.18, \(Y_{1}\) and \(Y_{2}\) denoted the lengths of life, in hundreds of hours, for components of types I and II, respectively, in an electronic system. The joint density of \(Y_{1}\) and \(Y_{2}\) is
\(f\left(y_{1}, y_{2}\right)=\left\{\begin{array}{ll}
(1 / 8) y_{1} e^{-\left(y_{1}+y_{2}\right) / 2}, & y_{1}>0, y_{2}>0, \\
0, & \text { elsewhere. }
\end{array}\right.
\)
One way to measure the relative efficiency of the two components is to compute the ratio \(Y_{2} / Y_{1}\). Find \(E\left(Y_{2} / Y_{1}\right)\). [Hint: In Exercise 5.61, we proved that \(Y_{1}\) and \(Y_{2}\) are independent.]
Equation Transcription:
Text Transcription:
Y_1
Y_2
Y_1
Y_2
f(y_1,y_2)={_0, elsewhere. ^(1/8)y_1e^-(y_1+y_2)/2, y_1>0,y_2>0,
Y_2/Y_1
E(Y_2/Y_1)
Y_1
Y_2
Questions & Answers
QUESTION:
In Exercise 5.18, \(Y_{1}\) and \(Y_{2}\) denoted the lengths of life, in hundreds of hours, for components of types I and II, respectively, in an electronic system. The joint density of \(Y_{1}\) and \(Y_{2}\) is
\(f\left(y_{1}, y_{2}\right)=\left\{\begin{array}{ll}
(1 / 8) y_{1} e^{-\left(y_{1}+y_{2}\right) / 2}, & y_{1}>0, y_{2}>0, \\
0, & \text { elsewhere. }
\end{array}\right.
\)
One way to measure the relative efficiency of the two components is to compute the ratio \(Y_{2} / Y_{1}\). Find \(E\left(Y_{2} / Y_{1}\right)\). [Hint: In Exercise 5.61, we proved that \(Y_{1}\) and \(Y_{2}\) are independent.]
Equation Transcription:
Text Transcription:
Y_1
Y_2
Y_1
Y_2
f(y_1,y_2)={_0, elsewhere. ^(1/8)y_1e^-(y_1+y_2)/2, y_1>0,y_2>0,
Y_2/Y_1
E(Y_2/Y_1)
Y_1
Y_2
ANSWER:
Answer:
Step 1 of 1:
In Exercise
and
denoted the lengths of life, in hundreds of hours, for components of types
respectively, in an electronic system.
The joint probability density function of and
is given by,
One way to measure the relative efficiency of the two components is to compute the ratio
We need to find the value of
The marginal density functions of and
respectively, are given by
Hence the marginal density functions for is,