# Ratio of two random variables squared

#### TimC

##### New Member
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

Related but not directly the answer is that the ratio of two independent standard normal random variables is Cauchy distributed

#### Buckeye

##### Member
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)

#### Buckeye

##### Member
So, it would be F (1,1) I think.

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#### TimC

##### New Member
So, it would be F (1,1) I think.
Buckeye, many thanks for the steer to the F distribution function. I think this may be the one. The variables X and Y have different variances, do you know how that is incorporated into the function?

thanks

#### TimC

##### New Member
Related but not directly the answer is that the ratio of two independent standard normal random variables is Cauchy distributed
Dason, thanks for the input

#### Dason

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
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