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Probability and Statistics


Cumulative Distribution Function

Definition:(Cumulative distribution function)
Let XX be a random variable defined on (Ω,F,P)(\Omega,\mathscr{F},P), then the function FX:RRF_X:\mathbb{R}\to\mathbb{R},

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

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

To study probability distribution PXP^X of XX, it is suffices to study the CDFCDF FXF_X of XX, as PXP^X is completely determined by values of FXF_X