Marginal PDFs for Joint PDF of X and Y

[countzander]

Homework Statement

Suppose that ∫_X,Y(x,y) = λ²e^-λ(x+y), 0 ≤ x, 0 ≤ y

Find E(X + Y)

Homework Equations

E(X + Y) = E(X) + E(Y)

The Attempt at a Solution

Since the expected vale of a sum is the sum of the expected values, I attempted to find the marginal pdfs of the joint pdf. But when calculating the integral for the marginal probability of X, p_X(x) = ∫λ²e^-λ(x+y) dy from 0 to ∞, the result is an undefined statement, division by zero.

 

[Ray Vickson]

Homework Statement

Suppose that ∫_X,Y(x,y) = λ²e^-λ(x+y), 0 ≤ x, 0 ≤ y

Find E(X + Y)

Homework Equations

E(X + Y) = E(X) + E(Y)

The Attempt at a Solution

Since the expected vale of a sum is the sum of the expected values, I attempted to find the marginal pdfs of the joint pdf. But when calculating the integral for the marginal probability of X, p_X(x) = ∫λ²e^-λ(x+y) dy from 0 to ∞, the result is an undefined statement, division by zero.

Remember that an integral from 0 to ∞ is a limit of the integral from 0 to U as U → ∞. Just do the integral from 0 to U first, then take the limit. Do it properly, and do it carefully.

 

[countzander]

That's what I did. As I said in the original post, the limit is 0. But because the 0 appears in the denominator, the integral is undefined.

Does anyone know where the problem is?

 

[Mark44]

That's what I did. As I said in the original post, the limit is 0. But because the 0 appears in the denominator, the integral is undefined.

I don't think so. Keep in mind that e^{-λ(x + y)} = e^-λx * e^-λy, and that you are integrating with respect to y.

Also, both e^x and e^-x are positive for all real numbers x, so I think you might be confused about 0 appearing in the denominator.

 

[Ray Vickson]

That's what I did. As I said in the original post, the limit is 0. But because the 0 appears in the denominator, the integral is undefined.

Does anyone know where the problem is?

You need to show us your work, step-by-step. Otherwise, there is no way we can help you.

 

[statdad]

"Since the expected vale of a sum is the sum of the expected values, I attempted to find the marginal pdfs of the joint pdf. "

There is no need to do that.

 

[Ray Vickson]

"Since the expected vale of a sum is the sum of the expected values, I attempted to find the marginal pdfs of the joint pdf. "

There is no need to do that.

Agreed. But he ought to be ABLE to do it if he wants to pass the course.