Probabilities with a Distribution Function

Topics: Distribution Function - Probability Measure - Probability

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(theorem)

Let $X$ be a random variable, $a\in \mathbb{R}$ and $F_X$ a distribution function.

From the definition of a distribution function at $a$ (denoted $F_X(a)$) and the definition of a distribution function at $a$ from the left (denoted $F_X(a_{-})$), we get that:

1. $\mathbb{P}(X\le a)=F_X(a)$
2. $\mathbb{P}(X>a)=1-F_X(a)$
3. $\mathbb{P}(X<a)=F_X(a_{-})$
4. $\mathbb{P}(X\ge a)=1-F_X(a_{-})$

And that:

1. $\mathbb{P}(a<X\le b)=F_X(b)-F_X(a)$
2. $\mathbb{P}(a\le X\le B)=F_X(b)-F_X(a_{-})$
3. $\mathbb{P}(a\le X<b)=F_X(b_{-})-F_X(a_{-})$
4. $\mathbb{P}(a<X<b)=F_X(b_{-})-F_X(a)$

Additionally:

1. $\mathbb{P}(X=a)=F_X(a)-F_X(a_{-})$

Note that the distribution function is continuous at $x$ if and only if $\mathbb{P}(X=x)=0$ .