(Probability/Statistics) Transformation of Bivariate Random Variable

[rayge]

Homework Statement

Let $X_1, X_2$ have the joint pdf $h(x_1, x_2) = 8x_1x_2, 0<x_1<x_2<1$, zero elsewhere. Find the joint pdf of $Y_1=X_1/X_2$ and $Y_2=X_2$.

Homework Equations

$$p_Y(y_1,y_2)=p_X[w_1(y_1,y_2),w_2(y_1,y_2)]$$ where $w_i$ is the inverse of $y_1=u_1(x_1,x_2)$

The Attempt at a Solution

We can get $X_1=Y_1Y_2$ and $X_2=Y_2$. Naively plugging in $y$, we can get $8y_1y_2^2$. However this isn't right according to the back of the book.

I thought it might have to do with finding the marginal distributions of $x_1, x_2$, but that doesn't seem to lead me anywhere either. Any thoughts welcome!

 

[Ray Vickson]

Homework Statement

Let $X_1, X_2$ have the joint pdf $h(x_1, x_2) = 8x_1x_2, 0<x_1<x_2<1$, zero elsewhere. Find the joint pdf of $Y_1=X_1/X_2$ and $Y_2=X_2$.

Homework Equations

$$p_Y(y_1,y_2)=p_X[w_1(y_1,y_2),w_2(y_1,y_2)]$$ where $w_i$ is the inverse of $y_1=u_1(x_1,x_2)$

The Attempt at a Solution

We can get $X_1=Y_1Y_2$ and $X_2=Y_2$. Naively plugging in $y$, we can get $8y_1y_2^2$. However this isn't right according to the back of the book.

I thought it might have to do with finding the marginal distributions of $x_1, x_2$, but that doesn't seem to lead me anywhere either. Any thoughts welcome!

You forgot the Jacobian, necessary to transform dx1*dx2 into h(y1,y2)*dy1*dy2

 

[rayge]

Thanks! That was it.