Solution Found!
The lifetimes, in months, of two components in a system,
Chapter 2, Problem 20E(choose chapter or problem)
The lifetimes, in months, of two components in a system, denoted and
, have joint probability density function
\(f(x)=\left\{\begin{array}{cc} 4 x y e^{-(2 x+y)} & x>0 \text { and } y>0 \\ 0 & \text { otherwise } \end{array}\right. \)
a. What is the probability that both components last longer than one month?
b. Find the marginal probability density functions \(f_X(x)\) and \(f_Y(y)\).
c. Are and
independent? Explain.
Equation Transcription:
Text Transcription:
f(x)={_0 otherwise ^4xye^{-(2x+y)} x>0 and y>0
f_X(x)
f_Y(y)
Questions & Answers
QUESTION:
The lifetimes, in months, of two components in a system, denoted and
, have joint probability density function
\(f(x)=\left\{\begin{array}{cc} 4 x y e^{-(2 x+y)} & x>0 \text { and } y>0 \\ 0 & \text { otherwise } \end{array}\right. \)
a. What is the probability that both components last longer than one month?
b. Find the marginal probability density functions \(f_X(x)\) and \(f_Y(y)\).
c. Are and
independent? Explain.
Equation Transcription:
Text Transcription:
f(x)={_0 otherwise ^4xye^{-(2x+y)} x>0 and y>0
f_X(x)
f_Y(y)
ANSWER:
Solution 20E
Step1 of 4:
Let us consider a random variables X and Y they presents the lifetimes, in months, of two components in a system and they have joint probability density function.
Here our goal is:
a).We need to find the probability that both components last longer than one month?
b).We need to find the marginal probability density functions
c).We need to check whether X and Y independent? Explain.
Step2 of 4:
a).
First integrate above equation with respect to “y” we get
Apply integration by parts we get
Apply integral substitution and let u = -2x-y
du = -1dy
)
Substitute “u” value in above equation we get
Now,
Integrate above equation with respect x we get