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?
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I fear you might be confusing two concepts that use the same letter for totally different purposes.
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.