MA2102

Probability and Statistics

Lecture-10

Cumulative Distribution Function

Definition:(Cumulative distribution function)
Let $X$ be a random variable defined on $(\Omega,\mathscr{F},P)$, then the function $F_X:\mathbb{R}\to\mathbb{R}$,

defined by, F_X(x)=P^X(-\infty,x]$$=$P(X\le x)$$=$$P(\{\omega\in\Omega:X(\omega)\le x\})$
is called Cumulative distribution function($CDF$) of $X$

We are not so good with dealing with set functions(like $P^X$), but we are good with dealing with functions of real variable(like $F_X$) thanks to Calculus.

To study probability distribution $P^X$ of $X$, it is suffices to study the $CDF$ $F_X$ of $X$, as $P^X$ is completely determined by values of $F_X$