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?

  • 2
    $\begingroup$ Not having these differences is close to being the definition of a continuous distribution function $\endgroup$
    – Henry
    Nov 30 '19 at 13:10

$P(X=a)=0$ 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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.