# F-Distribution theoretical question

Given a sample $$X_1,...,X_N$$. If $$X_i \overset{idd}{\sim} F$$ which will be the distribution of sample $$Y_1, ..., Y_N$$ where $$Y_i=F(X_i)$$ and why?

An $$F$$ distribution is a particular continuous distribution. It arises in ANOVA and linear regression.
$$F$$ as a function typically denotes the cumulative distribution function (CDF) for some random variable $$X$$, where it tends to be denoted $$F_X$$.
Applying the CDF as a transformation transforms a continuous random variable to $$U(0,1)$$. That applies for any continuous distribution, not just $$F$$-distributions. Consequently, if you have a continuous distribution and transform it according to its CDF, your transformed variable has a $$U(0,1)$$ distribution.
That is, $$Y_i=F_{X_i}(X_i)\sim U(0,1)$$. This is called the probability integral transform.