For continuous random variables X and Y with joint

Chapter 2, Problem 16E

(choose chapter or problem)

Get Unlimited Answers
QUESTION:

For continuous random variables X and Y with joint probability density function

\(f(x)=\left\{\begin{array}{cc}

x e^{-(x+x y)} & x>0 \text { and } y>0 \\

0 & \text { otherwise }

\end{array}\right.

\)

a. Find \(P(X>1 \text { and }Y>1)\).

b. Find the marginal probability density functions \(f_X(x)\) and \(f_Y(y)\).

c. Are X and Y independent? Explain.

Equation Transcription:

Text Transcription:

f(x)={_0              otherwise ^xe^-(x+xy)      x>0 and y>0

P(X>1 and Y>1)

fX(x)

fY(y)

Questions & Answers

QUESTION:

For continuous random variables X and Y with joint probability density function

\(f(x)=\left\{\begin{array}{cc}

x e^{-(x+x y)} & x>0 \text { and } y>0 \\

0 & \text { otherwise }

\end{array}\right.

\)

a. Find \(P(X>1 \text { and }Y>1)\).

b. Find the marginal probability density functions \(f_X(x)\) and \(f_Y(y)\).

c. Are X and Y independent? Explain.

Equation Transcription:

Text Transcription:

f(x)={_0              otherwise ^xe^-(x+xy)      x>0 and y>0

P(X>1 and Y>1)

fX(x)

fY(y)

ANSWER:

Solution 16E

Step1 of 3:

We have continuous random variables X and Y with joint probability density function

Here our goal is:

a).We need to find P(X > 1 and Y > 1).

b).We need to find the marginal probability density functions  

c).We need to check Are X and Y independent? Explain.

Step2 of 3:

a).

Consider,

 

                                                

Integrate above equation with respect to “y” we get

                                                

                                 

                                 =    

                                    = 0 - (-)

                           =

Where “e” is mathematical constant and its value is approximately 2.71828. Substitute “e” i above equation we get

                                     

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