0

In Wikipedia appears the pdf's equation for $XY$ and $X/Y$, where $X$ and $Y$ are given independent random variables. The equations are

For product $Z=XY$

$$f_Z(z)=\displaystyle\int_{-\infty}^{\infty}f_X(x)f_Y\left(\frac{z}{x}\right)\frac{1}{|x|}dx$$

and for the quotient $Z=X/Y$

$$f_Z(z)=\displaystyle\int_{-\infty}^{\infty}|y|f_{XY}(zy,y)dy$$

This equation appears in the following two links:

http://en.wikipedia.org/wiki/Ratio_distribution#Derivation

http://en.wikipedia.org/wiki/Product_distribution#Derivation_for_independent_random_variables

Why this two equation holds?

Thanks!

EQJ
  • 4,369

1 Answers1

1

You need to know Jacobian and change of variables. I will do the quotient, product is almost identical.

Let us define the following transformation $Z=X/Y$ and $T=Y$. We are going to find joint density $p(Z,T)$ by change of variables from $X,Y$ to $Z,T$. So inverse transformation is, $$ X=YT\\ Y=T $$ and Jacobian of this transformation is given by, $$ J=\begin{vmatrix} \delta X/\delta Z & \delta X/\delta T \\ \delta T/\delta Z & \delta Y/\delta T \end{vmatrix} = \begin{vmatrix} T & Z \\ 0 & 1 \end{vmatrix} =|T| $$ Then, $$ p(Z,T)=p(X,Y) J=p(X)p(Y)|T|=p_X(ZT)p_Y(T)|T| $$ We are looking for $Z$ so we can marginalize out $T$ which gives, $$ p(Z)=\int p_{X,Y}(ZT,T)|T|dT=\int p_{X,Y}(ZY,Y)|Y|dY $$

To learn more check this document.

user64066
  • 2,480