Distribution Function From the Left

Topics: Distribution Function - Probability

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(definition)

In Distribution Function, we defined what $F_X(x)$ is; we defined a distribution function $F_X$ exactly at $x\in \mathbb{R}$.

Now, we’ll define $F_X$ at $x$ from the left, denoted $F_X(x_{-})$, as:

$$F_X(x_{-}):=\underset{y\to x}{\lim }{F}_X(y)$$

(theorem)

Notice that:

$$F_X(x_{-})=\mathbb{P}(X<x)$$

…which is very similar to the normal definition of $F_X(x)$, but with $<$ instead of $\le$. As such, it is common to find $F_x(x)$ expressed in this form.