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?