Solution Found!
Let F1(y1) and F2(y2) be two distribution functions. For
Chapter 5, Problem 66E(choose chapter or problem)
Let \(F_{1}\left(y_{1}\right)\) and \(F_{2}\left(y_{2}\right)\) be two distribution functions. For any \(\alpha,-1 \leq \alpha \leq 1\), consider \(Y_{1}\) and \(Y_{2}\) with joint distribution function
\(F\left(y_{1}, y_{2}\right)=F_{1}\left(y_{1}\right) F_{2}\left(y_{2}\right)\left[1-\alpha\left\{1-F_{1}\left(y_{1}\right)\right\}\left\{1-F_{2}\left(y_{2}\right)\right\}\right] .\)
a What is \(F\left(y_{1}, \infty\right)\), the marginal distribution function of \(Y_{1}\) ? [Hint: What is \(F_{2}(\infty)\)?]
b What is the marginal distribution function of \(Y_{2}\)?
c If \(\alpha=0\) why are \(Y_{1}\) and \(Y_{2}\) independent?
d Are \(Y_{1}\) and \(Y_{2}\) independent if \(\alpha \neq 0\)? Why?
Equation Transcription:
Text Transcription:
F_1(y_1)
F_2(y_2)
alpha,-1</=alpha</=1
Y_1
Y_2
F(y_1,y_2)=F_1(y_1)F_2(y_2)[1-alpha{1-F_1(y_1)}{1-F_2(y_2)}].
F(y_1,infinity)
Y_1
F_2(infinity)
Y_2
alpha=0
Y_1
Y_2
Y_1
Y_2
Alpha not = 0
Questions & Answers
QUESTION:
Let \(F_{1}\left(y_{1}\right)\) and \(F_{2}\left(y_{2}\right)\) be two distribution functions. For any \(\alpha,-1 \leq \alpha \leq 1\), consider \(Y_{1}\) and \(Y_{2}\) with joint distribution function
\(F\left(y_{1}, y_{2}\right)=F_{1}\left(y_{1}\right) F_{2}\left(y_{2}\right)\left[1-\alpha\left\{1-F_{1}\left(y_{1}\right)\right\}\left\{1-F_{2}\left(y_{2}\right)\right\}\right] .\)
a What is \(F\left(y_{1}, \infty\right)\), the marginal distribution function of \(Y_{1}\) ? [Hint: What is \(F_{2}(\infty)\)?]
b What is the marginal distribution function of \(Y_{2}\)?
c If \(\alpha=0\) why are \(Y_{1}\) and \(Y_{2}\) independent?
d Are \(Y_{1}\) and \(Y_{2}\) independent if \(\alpha \neq 0\)? Why?
Equation Transcription:
Text Transcription:
F_1(y_1)
F_2(y_2)
alpha,-1</=alpha</=1
Y_1
Y_2
F(y_1,y_2)=F_1(y_1)F_2(y_2)[1-alpha{1-F_1(y_1)}{1-F_2(y_2)}].
F(y_1,infinity)
Y_1
F_2(infinity)
Y_2
alpha=0
Y_1
Y_2
Y_1
Y_2
Alpha not = 0
ANSWER:
Solution
Step 1 of 4
a) We have to write the marginal distribution function of Y1
Given that
Now
=
[Since ]
=
=is a marginal distribution of Y1
Hence is the marginal distribution function of Y1