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Let $F_X$ denote cumulative distribution function and $f_X$ denote probability density function.

If $X$ is a continuous random variable, then the following holds true:

$$\begin{align*}P(a \leq X < b) = P(a < X \leq b) = P(a < X < b) = P(a \leq X \leq b) &= \int_{a}^{b} f_X(x)dx \\ &= F_X(b) - F_X(a)\end{align*}$$

How can I easily explain that the used inequality symbols $\{<, \leq, >, \geq \}$ don't matter? Don't they?

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    $\begingroup$ Not having these differences is close to being the definition of a continuous distribution function $\endgroup$
    – Henry
    Nov 30 '19 at 13:10
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$P(X=a)=0$ for any $a$. Note that $[a,b]\setminus [a,b)=\{b\}$ so $P(a\leq X \leq b) -P(a\leq X <b)=P(X=b)=0$. Similar argument holds for other cases.

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