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:
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.