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What is the difference between a probability and a probability density function?

$\bullet$ Is it true that "in a probability density function, the area under the curve tells you the probability"? So consider a Gaussian probability density function, $f(x)=\frac{1}{\sigma\sqrt{2\pi}}e^{-\frac{(x-\mu)^{2}}{2\sigma^{2}}}$, now what is the probability of $x$ be equal to $\mu$? The area under a point in a curve is zero, isn't it?

$\bullet$ What does it mean when one says "the normal distribution is a probability distribution", so the total area under the normal curve is 1? Do we have "distributions" that are not a probability distribution?

Reference: Normal Distribution | Examples, Formulas, & Uses - Scribbr

I'm trying to understand this paper[Oded Regev. "On lattices, learning with errors, random linear codes, and cryptography", in Journal of ACM, 2009. Section 2. Preliminaries, pages 12-15].

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  • $\begingroup$ What does it mean when one says "the normal distribution is a probability distribution", so the total area under the normal curve is 1? ...Who says this, as I have never heard this before. $\endgroup$
    – Andrew
    Commented Jul 30, 2023 at 20:33
  • $\begingroup$ @AndrewZhang In the reference: Normal Distribution | Examples, Formulas, & Uses - Scribbr. $\endgroup$ Commented Jul 31, 2023 at 18:06

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Yes. Everything you say is correct. Probability measure would be a more technically correct term. There are general measures which don’t sum up to 1 - you normally normalize them, dividing by the total, so it does sum to 1. If the integral is infinite, you can’t define a probability (e.g. the Lebesgue measure has density 1 on all real numbers, but since it’s total is infinite, it’s impossible to describe a uniformly chosen random real number and trying to use one leads to paradoxes).

There is a complicated more general notion of distribution but it doesn’t really generalize probability and is more useful in advanced analysis.

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