Let F1(y1) and F2(y2) be two distribution functions. For

Chapter 5, Problem 66E

(choose chapter or problem)

Get Unlimited 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

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 


Add to cart


Study Tools You Might Need

Not The Solution You Need? Search for Your Answer Here:

×

Login

Login or Sign up for access to all of our study tools and educational content!

Forgot password?
Register Now

×

Register

Sign up for access to all content on our site!

Or login if you already have an account

×

Reset password

If you have an active account we’ll send you an e-mail for password recovery

Or login if you have your password back