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Suppose that the joint probability density function of X1 and X2 is given by: $$f(x_1; x_2) = exp(−x_1 − x_2)$$ $x_1 > 0$; $x_2 > 0$ and $0$ $elsewhere$. We define Y1 and Y2 as follows: $$Y_1 = X_1 + X_2$$ and $$Y_2= \frac{X_1}{X_1+X_2}$$

Determine the distribution of Y2.

SOLUTION:

I have used transformation theorem to find the distribution of Y2. Firstly I found that $X_1=Y_2Y_2$ and $X_2=Y_1-Y_2Y_1$

Jacobian determinant yields $J = Y_1$

Finally, I get:

$$f(y_1,y_2) = e^{y_1}\times y_1$$

QUESTION: How are the boundaries for the new joint distribution determined? I always have troubles to determine the boundaries/range in this kind of examples. Is there a technique?

To determine the marginal distribution I "integrate out" the $y_1$ (assimung the boudary is $0<y_1<1$) as follows:

$$ f(y_2) = \int_0^{1} e^{y_1}\times y_1 dy_1 = [(y_1-1)e^{y^1}]_0^1=-1$$

This doesnt seem right. I guess the boundry is where i am making the mistake?

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Should be $e^{-y_1}$ and irrespective of that your evaluation of the integral can't be right as it has to be positive. Anyway, there is no $y_2$ in the density, so it must be uniform on $[0,1]$.

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