Probability Distribution Function

Definition (PDF)

Let $X\in \mathbb{R}$ be a random variable. The Probability Distribution Function (PDF) denoted as $f_X$ or $f$ is defined by

$$f_X(x)=F_X^{\prime }(x)$$

the derivative of the CDF apart from a “few points”. Some properties are that

• $f_X\ge 0$
• ${\int }_{-\infty }^{\infty }{f}_X(x)dx=1$
• $P(a\le X\le b)=F_X(b)-F_X(a)={\int }_a^{b}f(x)dx$
• $P(X=x)=F_X(x)-F_X(x^{-})=F_X(x)-{\lim }_{y\to x}{F}_X(y)$.
  • This is particularly useful for discrete random variables. We call this a Probability Mass Function, or a PMF.

Notation

To denote that a RV $X$ has a particular distribution will be by

$$X\sim fX\sim F$$

such that $X$ has PDF $f$ and CDF $F$.