Let X and Y be normal random variables with mean 0, variance $\sigma^2$ and correlation coefficient $\rho \in (-1,1)$, so that the density is given by

$$f(x,y) = \cfrac{1}{2\pi \sigma^2\sqrt{a-\rho^2}}\,\,\exp{\{-\cfrac{1}{2\sigma^2(1-\rho^2)}[x^2 - 2\rho xy+y^2]\}}$$

How to determine the distribution of $Z=\cfrac{X}{Y}$ ?


closed as off-topic by heropup, colormegone, Leucippus, Shailesh, choco_addicted Apr 10 '16 at 1:53

This question appears to be off-topic. The users who voted to close gave this specific reason:

  • "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – heropup, colormegone, Leucippus, Shailesh, choco_addicted
If this question can be reworded to fit the rules in the help center, please edit the question.

  • $\begingroup$ Do you know how you in general get the pdf of a random variable $Z=X/Y$ when you have the joint pdf $f_{XY}(x,y)$? $\endgroup$ – JKnecht Apr 9 '16 at 19:49
  • $\begingroup$ Yes, there is the specific formula $f_Z(z)=\int_{-\infty}^{+\infty}{|y| f_X(z y)f_Y(y)dy}$ $\endgroup$ – mic Apr 9 '16 at 19:55
  • $\begingroup$ But how do I find marginals of $f_{X,Y}$? I don't know how to integrate analytically a normal pdf. $\endgroup$ – mic Apr 9 '16 at 20:01

Let $Z=X/Y$ and $W=Y$.

The transformation $z=x/y,\: w=y$ has the inverse transformation

$x=zw, \: y=w$, and

$$ \bar{J}(z,w) =\begin{vmatrix} \frac{\partial{x}}{\partial{z}} & \frac{\partial{x}}{\partial{w}} \\ \frac{\partial{y}}{\partial{z}} & \frac{\partial{y}}{\partial{w}} \\ \end{vmatrix} $$

$$ \bar{J}(z,w)=\begin{vmatrix} w & z \\ 0 & 1 \\ \end{vmatrix} = w $$


$$f_{ZW}(z,w) = |w|f_{XY}(zw,w)$$

and the marginal pdf of $Z$ is

$$f_{Z}(z) = \int_{-\infty}^{\infty}|w|f_{XY}(zw,w)dw$$

  • $\begingroup$ Perfect, I got $f_Z(z) = \cfrac{1}{z^2 - 2\rho z +1}$. $\endgroup$ – mic Apr 9 '16 at 20:36

Not the answer you're looking for? Browse other questions tagged or ask your own question.