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Assume I have a continuous variable X with PDF f(x).

I know that, for every value of x, P(X=x) = 0, so the value f(x) is not the probability. But what is it exactly?

If we have f(a) = 2 * f(b), can we say that the value of a have twice of probability to appear compared to the value of b?

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  • $\begingroup$ Not really. I know that f(x) can >1. What I want to know is, for instance, f(2) =4 and f(3)=2, how could we interpret that? Can we say 2 is double of 3 in term of chance to appear? $\endgroup$ Sep 15 at 10:16

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You should start with the Can a probability distribution value exceeding 1 be OK? thread that explains the concept of probability density in detail.

If we have f(a) = 2 * f(b), can we say that the value of a have twice of probability to appear compared to the value of b?

You have twice the probability density for the second point. As you can learn from the mentioned thread, the probability density is probability per foot, so it is measured in the units relative to the units of your variable. It's closely related to the probability since you can ask about the probabilities by taking the integral

$$ \int_{a-\varepsilon}^{a+\varepsilon} f(x) dx = 2 \int_{b-\varepsilon}^{b+\varepsilon} f(x) dx $$

and now we are comparing probabilities of the points within the $\pm \varepsilon$ ranges around those values. Here indeed, the probability is two times higher. To convince yourself that they are quite similar, consider the scenario with $\varepsilon \to 0$. Saying it differently, probability densities, the same as probabilities, tell you relatively how "likely" different values are compared to each other. So the interpretation does not differ that much, but probabilities and probability densities are not the same and should not be confused.

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