Solution Found!
For continuous random variables X and Y with joint
Chapter 2, Problem 16E(choose chapter or problem)
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