Marginal Probability Distribution

[Rifscape]

Homework Statement

Two components of a laptop computer have the following joint probability density function for their useful lifetimes X and Y (in years):

f(xy)=xe^(−x(1+y)) 0 <= x <= y

0 otherwise

Find the marginal probability density function of X, fX(x). Enter a formula below. Use * for multiplication, / for division, ^ for power and exp for exponential function. For example, 3x^3*exp(-x/3) means 3x^3e^(-x/3).

I found the answer to this, it is e^(-x).

Find the marginal probability density function of Y, fY(y). Enter a formula below.

I found the answer to this one too, its 1/(1 + y)^2 .

What is the probability that the lifetime of at least one component exceeds 1 year (when the manufacturer's warranty expires)? Round your answer to 4 decimal places.

This is the part I'm having trouble on, I'm not really sure how to start or set up this question.

Thanks for the help.

Homework Equations

The marginal probability equations, I'm not sure how to write them here.

The Attempt at a Solution

I don't really know how to set up the third part.

 

[Ray Vickson]

Homework Statement

Two components of a laptop computer have the following joint probability density function for their useful lifetimes X and Y (in years):

f(xy)=xe^(−x(1+y)) 0 <= x <= y

0 otherwise

Find the marginal probability density function of X, fX(x). Enter a formula below. Use * for multiplication, / for division, ^ for power and exp for exponential function. For example, 3x^3*exp(-x/3) means 3x^3e^(-x/3).

I found the answer to this, it is e^(-x).

Find the marginal probability density function of Y, fY(y). Enter a formula below.

I found the answer to this one too, its 1/(1 + y)^2 .

What is the probability that the lifetime of at least one component exceeds 1 year (when the manufacturer's warranty expires)? Round your answer to 4 decimal places.

This is the part I'm having trouble on, I'm not really sure how to start or set up this question.

Thanks for the help.

Homework Equations

The marginal probability equations, I'm not sure how to write them here.

The Attempt at a Solution

I don't really know how to set up the third part.

The complement of the event "at least one component has a lifetime of >= 1 year" is "both components have lifetimes < 1 year".

 

[Rifscape]

The complement of the event "at least one component has a lifetime of >= 1 year" is "both components have lifetimes < 1 year".

Alright yeah that makes sense, the problem I have is how to set it up. Would I just do the double integral from 0 to 1 of the function with respect to x and y and then subtract?

 

[Rifscape]

The complement of the event "at least one component has a lifetime of >= 1 year" is "both components have lifetimes < 1 year".

Actually nevermind I got it, thanks for the help!