Does $Pr(F_X(x) < t) = Pr(x < F^{-1}(t))$?

Question

Consider a random variable $X$ with CDF $F_X(x) = Pr(X \leq x)$. I am wondering if $Pr(F_X(x) < t) = Pr(x < F^{-1}(t))$ holds in general for any $t$?

Answer 1

The comments have already explained that the answer is yes, as long as $0<t<1$ and $F_X$ has an inverse. More generally, most events $A$ that you have to work with in computational problems can be interpreted as logical statements, and $P\left(A\right)$ can be viewed as the probability that the statement defining $A$ is true. For example, $\left\{F_X \left( x \right) < t\right\}$ is a logical statement.

Then, to show that $P\left(A\right) = P\left(B\right)$, you have to argue that $A \Leftrightarrow B$ in the logical sense, so $A$ implies $B$ and $B$ implies $A$. That is certainly true in this case as long as $F_X$ is invertible since applying a monotonic function (in this case, $F_X^{-1}$) to both sides of an inequality preserves the inequality, so

$$\left\{F_X\left(x\right) < t\right\} \qquad \Leftrightarrow \qquad \left\{x <F_X^{-1}\left(t\right) \right\}.$$