# Probability Density Function vs. Probability

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?

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].

• 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. Commented Jul 30, 2023 at 20:33
• @AndrewZhang In the reference: Normal Distribution | Examples, Formulas, & Uses - Scribbr. Commented Jul 31, 2023 at 18:06