Marginal PDF integral not converging

Question

If I have the following marginal PDF of independent random varibales:

$p_X(x)=1 , 0<x<1$

$p_Y(y)=\frac{1}{2} , -1<y<1$

To get the joint PDF of:

W = XY

Z = X

I can use the following:

$p_{W,Z}(w,z)=p_X(g^{-1}(w,z)) p_Y(h^{-1}(w,z))|det(\frac{\partial(x,y)}{\partial(w,z)})|$

and get that $p_{W,Z}(w,z)=\frac{1}{2z}$

But now if I want to get the marginal PDF of W I can't find the correct limit for the integral and its not converging.

Answer 1

The $(x,y)$-domain is defined by the inequalities $$ 0\lt x\lt1,\qquad -1\lt y\lt1, $$ and the transformation is $$ x=z,\qquad y=w/z, $$ hence the $(w,z)$-domain is defined by the inequalities $$ 0\lt z\lt1,\qquad -1\lt w/z\lt1, $$ or, equivalently, $$ 0\lt z\lt1,\qquad-z\lt w\lt z. $$ This is a good example of the reason why one should always mention the domain in the densities, in the present case, the joint density $f_{W,Z}$ is not what you write but $$ f_{W,Z}(w,z)=\frac1{2z}\,\mathbf 1_{|w|\lt z\lt1}. $$ Hence, $$ f_W(w)=\int_\mathbb Rf_{W,Z}(w,z)\mathrm dz=\mathbf 1_{|w|\lt1}\int_{|w|}^1\frac1{2z}\mathrm dz, $$ that is, $$ f_W(w)=-\frac12\log|w|\cdot\mathbf 1_{|w|\lt1}. $$

Answer 2

As an alternative to Did's answer, consider that $W = XY$ takes on values in $(-1,1)$, and that for any $w, 0 < w < 1$, and so $$P\{W > w\} = P\{XY > w\} = \int_{x=w}^1\int_{y=\frac wx}^1 \frac 12\, \mathrm dy \, \mathrm dx = \left.\left.\frac 12\right[1 - w + w\ln w\right].$$ Thus, for $0 < w < 1$, $f_w(w) = -\left.\left.\frac{\mathrm d}{\mathrm dw}\right[1-F_W(w)\right] = -\frac{\mathrm d}{\mathrm dw} P\{W > w\} = -\frac 12 \ln w$.

Now, since $Y$ and $-Y$ have the same distribution (and thus $f_Y(y)$ is an even function of $y$), $W=XY$ and $-W = -XY = X(-Y)$ also have the same distribution and so $f_W(w)$ is an even function of $w$. This gives us that $$f_W(w) = \left(-\frac 12 \ln |w|\right)\cdot \mathbf 1_{|w| < 1}.$$ even