Probability Density Function (pdf)

Definition

Let $X$ : Continuous random variable

If $f_X(x)=P[X=x]=0,\forall x\in \mathbb{R}$

Then we say that $f_X(x)$ is the probability density function (pdf) of $X$

Properties

• $F_X(x)={\int }_{-\infty }^x{f}_X(t)dt$
• $\frac{d}{dx}{F}_X(x)=f_X(x)$
• $P(a<X\le b)=F_X(b)-F_X(a)={\int }_a^b{f}_X(t)dt$
• $P(a<X\le b)=P(a\le X\le b)=P(a\le X<b)=P(a<X<b)$

Additionally, pdfs satisfies the following properties:

1. $f_X(x)\ge 0$
2. ${\int }_{-\infty }^{\infty }{f}_X(t)dt=1$