Density function

Definition

The Radon–Nikodym Derivative of a Distribution Function

A density function $f_X$ is a function for the Random Variable $X$ on a Probability Space $(\Omega ,\mathcal{F},P)$ satisfying $F_X(x)={\int }_{-\infty }^x{f}_X(y)dy$

Induced Dessity Function

$P_X(x)=P(X=x):=P(\{\omega :X(\omega )=x\})=P(X^{-1}(x))$ where we call the $P_X$ an induced density function by the Random Variable $X$

Facts

$\forall x_0\in \mathbb{R},P(X=x_0)=F_X(x_0)-\underset{x\to x_0}{\lim }{F}_X(x)$

$P(a<X\le b)=F_X(b)-F_X(a)={\int }_a^b{f}_X(t)dt$