- #1

Yagoda

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## Homework Statement

Two RVs X

_{1}and X

_{2}are continuous and have joint pdf [itex]

f_{X_1,X_2}(x_1, x_2) = \begin{cases} x_1+x_2 &\mbox{for } 0 < x_1 < 1; 0 < x_2 < 1

\\

0 & \mbox{ } \text{otherwise}. \end{cases} [/itex]

Find the pdf of [itex]Y = X_1X_2[/itex].

## Homework Equations

I'm using the transformation "shortcut' that says if U = g1(x,y), V = g2(x,y) and h1(u,v) = x, h2(u,v) = y then

[itex]f_{U,V}(u,v) = f_{X,Y}(h_1(u,v), h_2(u,v) \times |J| [/itex] where |J| is the absolute value of the determinant of the Jacobian of h1 and h2.

## The Attempt at a Solution

In this setting I let [itex]Y = X_1X_2[/itex] and [itex]Z = X_1[/itex] so [itex]h_1(y,z) = Z[/itex] and [itex]h_2(y,z) = Y/Z[/itex]. Then I calculated [itex]|J| = \frac{1}{Z}[/itex] so plugging in I got that

[itex]f_{Y,Z}(y,z) = (z + \frac{y}{z})(\frac{1}{z}) = 1 + \frac{y}{z^2}[/itex] where [itex]0 < z < 1, 0 < y < z[/itex].

So to find the pdf of Y, integrate with respect to z over all possible values of z, but this integral does not converge when evaluated from 0 to 1.

I've tried other choices for Z, but I've ended up with the same problem.

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