Upon considering this definition:
" A random variable is said to be continuous if and only if:
- its cumulative distribution function $F_X: \mathbb{R}\to [0,1]$ is a continuous function.
- there is a non-negative function $f_X:\mathbb{R} \to [0,\infty)$ such that $$F_X(x)=\int_{-\infty}^{x}f_X(t)dt\space\space\forall x \in \mathbb{R}$$
the function $f_X$ is called the probability density function of the continuous random variable X."
1) I seem to have the idea that the pdf associates infinitesimally small points in in the range of X with probability masses i.e. as we integrate over $R_X$ we pick up all the probability masses of the infinites points in $R_X$ (as X is continuous)
Is this intuition correct?
If not, what is the "correct" intuition for what probability density functions "are" / what they "do"?
If a random variable $X$ on a probability space $(\Omega, \mathcal F, \mathbb P)$ is continuous, then its distribution function $F(x):= \mathbb P(X\leqslant x)$ is absolutely continuous, and thus defines a measure $\mu_F$ on $\mathbb R$ with $\mu_F((-\infty,x])=F(x)$ such that $\mu_F\ll\mathsf m$, where $\mathsf m$ is Lebesgue measure. Because both $\mu$ and $\mathsf m$ are $\sigma$-finite, by the Radon-Nikodym theorem there exists a measurable function $f:\mathbb R\to[0,\infty)$ such that for any event $E\in\mathcal F$, $$\mu_F(A) = \int_E f\ d\mathsf m. $$ This function $f$ is called the density of $X$.