4.2.3 Obtaining f(x) from F(x)

For $X$ discrete, the pmf is obtained from the cdf by taking the difference between two $F\left(x\right)$ values. The continuous analog of a difference is a derivative. The following result is a consequence of the Fundamental Theorem of Calculus.

Proposition

If $X$ is a continuous rv with $\mathrm{pdf}f\left(x\right)$ and $\mathrm{cdf}F\left(x\right)$, then at every $x$ at which the derivative $F^{\prime }\left(x\right)$ exists, $F^{\prime }\left(x\right)=f\left(x\right).$

EX 4.8 EX 4.6 continued