Solution Found!
Solved: In Exercise 5.10, we proved that is a valid joint
Chapter 5, Problem 54E(choose chapter or problem)
In Exercise 5.10, we proved that
\(f\left(y_{1}, y_{2}\right)=\left\{\begin{array}{ll}
1, & 0 \leq y_{1} \leq 2,0 \leq y_{2} \leq 1,2 y_{2} \leq y_{1}, \\
0, & \text { elsewhere }
\end{array}\right.
\)
is a valid joint probability density function for \(Y_{1}\), the amount of pollutant per sample collected above the stack without the cleaning device, and \(Y_{2}\), the amount collected above the stack with the cleaner. Are the amounts of pollutants per sample collected with and without the cleaning device independent?
Equation Transcription:
Text Transcription:
f(y_1,y_2)={_0, elsewhere ^1, 0</=y_1</=2,0</=y_2</=1,2y_2</=y_1,
Y_1
Y_2
Questions & Answers
QUESTION:
In Exercise 5.10, we proved that
\(f\left(y_{1}, y_{2}\right)=\left\{\begin{array}{ll}
1, & 0 \leq y_{1} \leq 2,0 \leq y_{2} \leq 1,2 y_{2} \leq y_{1}, \\
0, & \text { elsewhere }
\end{array}\right.
\)
is a valid joint probability density function for \(Y_{1}\), the amount of pollutant per sample collected above the stack without the cleaning device, and \(Y_{2}\), the amount collected above the stack with the cleaner. Are the amounts of pollutants per sample collected with and without the cleaning device independent?
Equation Transcription:
Text Transcription:
f(y_1,y_2)={_0, elsewhere ^1, 0</=y_1</=2,0</=y_2</=1,2y_2</=y_1,
Y_1
Y_2
ANSWER:
Answer:
Step 1 of 1:
We have given the joint probability density function.
Is a valid joint probability density function for the amount of pollutant per sample collected above the stack without the cleaning device, and for
the amount collected above the stack with the cleaner.
Are the amounts of pollutants per sample collected with and without the cleaning device independent?
If are continuous random variables with joint density function
and marginal density functions respectively, then
are independent if and only if