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I have empirical data representing a CDF. I fitted a normal/gaussian CDF (the line) to it, call it F(x)

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I have found that squaring the data points gives a better fit; F(x)² = P(X ≤ x)²:

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And here's their corresponding PDFs:

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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.

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

1 Answer 1

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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.

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