# Normalizing the joint probability density

I computed the kernel estimators for the copula density for two random variables using:

library(kdecopula)

kde.fit <- kdecop(u)

As the values of density can be greater than one I was wondering if I can normalized values by the maximum magnitude and call it normalized joint probability density?

Here is the bivariate copula density and the normalized bivariate copula density

Bivarite copula density:

Normalized bivarite copula density:

Thanks in advance for any helps.

In literature, normalization means integrating to $$1$$, not having a max value equal to $$1$$. So, joint or univariate densities are already normalized. For the nomenclature, for the function you have, I think max-normalized joint density would be a better name for it.
However, what you do is just scaling your joint PDF so that it hits $$1$$ at its maximum. Since both in your new function and the original density, these values can't be associated with probabilities, I see no use in doing so. It's not as much different as than multiplying your density with e.g. $$5$$.
• It is worth adding that if a function $f(x,y)$ is a density, then it must fulfill such condition: $\int\int_{R^2} f(x,y) dxdy=1$, stating that the volume below the density surface must be equal to one (IIIrd axiom of the probability definition). So it can be normalized, but it will no longer be probability density. Feb 13, 2020 at 15:07