# What does the PDF of a CDF² mean?

I have empirical data representing a CDF. I fitted a normal/gaussian CDF (the line) to it, call it F(x) I have found that squaring the data points gives a better fit; F(x)² = P(X ≤ x)²: And here's their corresponding PDFs: Link to desmos with all of this

I understand what the purple PDF is, i.e. P(X = x), for a loosely fitted curve. But what does the green PDF represent?

My thoughts: Since a PDF is the derivative of a CDF, and the derivative of F(x)² = 2 F(x), the green curve could be 2P(X = x). But that would mean the area under its curve would have to add up to 2, and it clearly equals 1.

• Errr no, the derivative of $[F(x)]^2$ is $2F(x)f(x)$. Remember the chain rule for differentiation from Calculus 101? May 29 at 2:14
• If your model does not have theoretical justification, you can revise your model to reflect what you have found. May 29 at 2:50

If $$X$$ and $$Y$$ are independent random variables with the same CDF $$F(x)$$, then $$P(\max(X,Y) \leq \alpha) = P(X \leq \alpha, Y \leq \alpha) = P(X\leq \alpha)P(Y \leq \alpha) = [F(x)]^2.$$ Thus, $$[F(x)]^2$$ is the CDF of the maximum of two independent random variables with the same CDF $$F(x)$$. How this notion can be fitted into your model remains to be seen.