# Why $P(a \leq X < b) = P(a < X \leq b) = P(a < X < b) = P(a \leq X \leq b)$?

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?

• Not having these differences is close to being the definition of a continuous distribution function 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 . Similar argument holds for other cases.