>[!info] Definition >If the [[Cumulative Distribution Function|CMF]] of a **continuous** [[Random Variable|random variable]] is differentiable, the PDF is defined as$ f_X(x)=\frac{\mathrm{d}F_X(x)}{\mathrm{d}x}$ >[!brainwaves] Additional Information >Mathematically, the density is then only defined with regard to the [[Measure|measure]], in almost every application this is the [[Lebesgue Measure|Lebesgue measure]] ($dx =d \lambda(x)$). --- >[!success] Properties >- Relationship with [[Cumulative Distribution Function|CDF]]$F_X(x) = \int_{- \infty}^{x} f_X(\bar{x}) \mathrm{d} \bar{x}$ >- From density to probability$P(a < X < b) = \int^{b}_{a} f_X(\bar{x}) \mathrm{d} \bar{x}$ >- Probability of a single point$P(X=x)=0 \, \forall x$