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The joint distribution for the length of life of two
Chapter 6, Problem 31E(choose chapter or problem)
The joint distribution for the length of life of two different types of components operating in a system was given in Exercise by
\(f\left(y_{1}, y_{2}\right)=\left\{\begin{array}{ll} (1 / 8) y_{1} e^{-\left(v_{1}+y_{2}\right) / 2}, & y_{1}>0, y_{2}>0 \\ 0, & \text { elsewhere } \end{array}\right.\)
The relative efficiency of the two types of components is measured by \(U=Y_{2} / Y_{1}\). Find the probability density function for \(U\).
Equation Transcription:
{
Text Transcription:
f(y1,y2)= {(1/8)y_1e^-(v_1=y_2)/2, y_1 > 0, y_2 > 0 0, elsewhere
U=Y_2/Y_1
U
Questions & Answers
QUESTION:
The joint distribution for the length of life of two different types of components operating in a system was given in Exercise by
\(f\left(y_{1}, y_{2}\right)=\left\{\begin{array}{ll} (1 / 8) y_{1} e^{-\left(v_{1}+y_{2}\right) / 2}, & y_{1}>0, y_{2}>0 \\ 0, & \text { elsewhere } \end{array}\right.\)
The relative efficiency of the two types of components is measured by \(U=Y_{2} / Y_{1}\). Find the probability density function for \(U\).
Equation Transcription:
{
Text Transcription:
f(y1,y2)= {(1/8)y_1e^-(v_1=y_2)/2, y_1 > 0, y_2 > 0 0, elsewhere
U=Y_2/Y_1
U
ANSWER:
Solution:
Step 1 of 2:
It is given that and
are random variables denoting the life lengths of two different types of components.
The joint density function is given by
f(
The relative efficiency is given by U=.We have to find the probability density function of U.