Transforming Two Normal Random Variables into a Non-Central F Distribution

[zli034]

I don't know if this is possible or not, let's see if this is a fun problem.

Let X_1 and X_2 be 2 independent normal random variables. They have different means and variances, and they are independent. I want to have a function that inputs X_1 and X_2, and it has a F distribution with degree of freedom 1,1.

We know the ratio of 2 mean squared errors are F distributed from ANOVA. But only 2 variables is hard to have means squared errors. How about other functional forms can make these two normals into F?

 

[Stephen Tashi]

Your terminology for the inputs and outputs is somewhat ambiguous. There is a difference between "a random variable" and "a realization of a random variable" (i.e. a sample). If you input "a random variable X_1", the input would be distribution. If you input "a realization of a random variable X_1" then the input would be a single number.

 

[zli034]

(X_1^2+X_2^2)/(X_1-X_2)^2 is a non-central F random variable. But the non-central parameter is unknown.