Finding joint pdf of independent variables.

[cookiesyum]

Homework Statement

Suppose X₁ and X₂ are two independent gamma random variables, and X₁~Gamma(a₁, 1) and X₂~(a₂, 1).

a) Find the joint pdf of Y₁ = X₁ + X₂, and Y₂ = X₁/(X₁ + X₂).

b) Show that Y₁ and Y₂ are independent.

c) Find the marginal distributions of Y₁ and Y₂.

The Attempt at a Solution

a) 1/[Gamma(a₁)Gamma(a₂)] * (y₁y₂)^{a₁ - 1} * (y₁ - y₁y₂)^{a₂ - 1} * e^-y₁ * y₁

b) Show joint pdf = product of individual pdfs of y₁ and y₂. To do this you have to find the marginal distributions so...

c) To find the marginal distributions, integrate away the extra variable. So, for the pdf of y₁ the integral you have to evaluate is...

(e^-y₁y₁/(Gamma(a₁)Gamma(a₂)) * integral of (y₁y₂)^{a₁ - 1} * (y₁ - y₁y₂)^{a₂ - 1} dy₂ as y₂ goes from 0 to 1...

But this integral is pretty hard to do by hand (unless I'm missing a trick). So is there a better way to do this problem?

 

[mighty2000]

Thank you for your question. I would like to offer some suggestions for approaching this problem. First, it is important to understand the definitions and properties of gamma random variables. In this case, X1 and X2 are independent gamma random variables with different parameters. This means that they follow different gamma distributions, but their values are not related to each other.

To find the joint pdf of Y1 and Y2, we can use the transformation method. This involves finding the Jacobian of the transformation from (X1, X2) to (Y1, Y2). The joint pdf can then be expressed as the product of the individual pdfs of Y1 and Y2. This approach may be more efficient than trying to integrate the joint pdf from scratch.

To show that Y1 and Y2 are independent, we can use the definition of independence for random variables. This means that the joint pdf of Y1 and Y2 can be expressed as the product of their individual pdfs, which we have already found in part (a). This also follows from the fact that X1 and X2 are independent, and the transformation from (X1, X2) to (Y1, Y2) does not introduce any dependence between Y1 and Y2.

To find the marginal distributions of Y1 and Y2, we can integrate the joint pdf with respect to the other variable. For Y1, we can integrate with respect to Y2, and for Y2, we can integrate with respect to Y1. This will give us the marginal pdfs of Y1 and Y2, respectively. As you mentioned, the integrals may be difficult to evaluate by hand, so it may be helpful to use a computer program or numerical integration techniques to obtain the results.

I hope this helps you approach the problem in a more systematic and efficient manner. Good luck with your work!