Strict Inequality Using CDF

Question

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)

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. — Did, May 04 '14 at 18:38

score 3 · Answer 1 · answered May 04 '14 at 18:37

3

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

answered May 04 '14 at 18:37

Did

279,727

I'll keep in mind the corrections for the future. Thanks. – user147813 May 04 '14 at 20:41

Strict Inequality Using CDF

1 Answers1