Ratio of two random variables squared

TimC

New Member
#1
Dear all,

Does anyone know where i can find the PDF for the the following Z:

Z = (X^2) / (Y^2), X and Y are zero mean Gaussian variables with standard deviations sigma_x and signma_y (X, Y independent and uncorrelated).

I know that X^2 and Y^2 are chi-squared but i have not been able to find a PDF of their ratio

Also, is there a formula for:

E{Z} and Var{Z}.

Many thanks for any help

cheers

Tim
 

Dason

Ambassador to the humans
#2
Related but not directly the answer is that the ratio of two independent standard normal random variables is Cauchy distributed
 
#3
If you divide the numerator RV by its degrees of freedom (n) and the denominator RV by its degrees of freedom (d), the result is F (n,d)
 

Dason

Ambassador to the humans
#7
So a few fun facts: A Cauchy distribution is the same as a T distribution with 1 degree of freedom. If you square a T distribution with k degrees of freedom it has the same distribution as a F(1, k) distribution. So F(1,1) is the same as the square of a T with 1 degree of freedom which is the same as a Cauchy that has been squared.

With that fun stuff out of the way - you still need to deal with your variances since all of those fun facts come from standard normals. Luckily you already have mean 0 so all you need to do is divide your random variables by their standard deviations.

Z = (X^2) / (Y^2) = ( (X/sd_x) * sd_x)^2 / ( (Y/sd_y) * sd_y)^2 = ( (X/sd_x)^2 / (Y/sd_y)^2) * (sd_x^2/sd_y^2). And that first component has a F(1,1) distribution. So you're looking at a F(1,1) scaled by sd_x^2 / sd_y^2
 

TimC

New Member
#8
Dason and Buckeye. Thanks for your help. The scaling process seems to work - i confirmed using simulations. You have assisted me in making progress in this topic.

cheers