2
$\begingroup$

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?

$\endgroup$

1 Answer 1

5
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.