Distribution function (Cumulative distribution function)

Definition 11.7 (Cumulative distribution function, continuous case). For a continuous random variable $X$ with pdf $f(x)$, the distribution function is

Remarks (comparing the two cases).

(1) In the discrete case $f(a)=P(X=a)$ directly; in the continuous case $f(a)$ is not the probability that $X=a$ — indeed $P(X=a)=0$ for every $a$, by Definition 11.5.

(2) Passing from discrete to continuous simply replaces every sum by the corresponding integral. …