probability density function

A PDF is a probability distribution that describes the relative likelihood of a continuous random variable taking on values in different ranges.

It’s a model of a random process:
→ You can do hypothesis testing with data
→ You can estimate parameters (mean, std, …) with a small sample

The probability of a random variable falling within a range is the integral of the PDF over that range:

$$P(a\le x\le b)={\int }_a^{b}f(x)dx$$

→ Probabilities are only meaningful over intervals/ranges, not individual points.
→ The value of $f(x)$ can be greater than $1$, as it represents density not probability.
→ $P(X=x)=0$ for any single point $x$, as it represents an interval of width zero.

Example: For a standard normal distribution, $f(0)\approx 0.399$, but $P(X=0)=0$, however, $P(-1\le X\le 1)\approx 0.683$, representing the probability of falling within that interval.

So while taking sums of probabilities over individual points works in a discrete context (probability mass function):

$$P(x\in \{a,b,c\})=P(x=a)+P(x=b)+P(x=c)=\sum _{x\in \{a,b,c\}}P(x)$$

It doesn’t work in a continuous context:

$$\begin{array}{r}P(x\in [a,c])\cancel{=\sum _{x\in [a,c]}P(x)}={\int }_{x\in [a,c]}P(x)dx\end{array}$$