Probability Density Function

A function used to specify the probability of a RV falling within a particular range of values. Maps a RV into a probability from $[0,1]$ It measures the rate at which a given probability accumulates around $x$ of RV $X$

Definition

The PDF is the function $f_X(x)$ such that: $P(X\in [a,b])={\int }_a^b{f}_X(x)dx$

Properties

• $0\le f_X(x),\forall x\in \mathbb{R}$ (Can have $f(x)>1!$)
• ${\int }_{-\infty }^{+\infty }{f}_X(x)dx=1=P(-\infty <X<+\infty )=P(S)$
• $F_X(x)={\int }_{-\infty }^x{f}_X(u)du⟹f_X(x)=\frac{d}{dx}{F}_X(x)=F_X^{\prime }(x)$ (PDF $\leftrightarrow$ CDF)