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Considering the definition of cumulative distribution function: $$F_{x}(x)=P[X\le x]=\int_{- \infty}^{x} f_{x}(x)dx$$ where $f_{x}$ is the probability density function of $x$, how can one obtain $P[X< x]$ ? (Note the strict inequality)

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  • $\begingroup$ Please use f_X and F_X, not f_x and F_x. Please do not use x both for the upper bound and for the argument of the integral. $\endgroup$
    – Did
    Commented May 4, 2014 at 18:38

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If the random variable $X$ has a PDF, then, for every $x$, $P(X=x)=0$ hence $P(X\lt x)=P(X\leqslant x)=F_X(x)$.

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  • $\begingroup$ I'll keep in mind the corrections for the future. Thanks. $\endgroup$
    – user147813
    Commented May 4, 2014 at 20:41

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