Properties of Distribution Functions

In the setting of Probability Spaces: Notation, let $\mu$ be a probability measure on $(\mathbb{R},\mathcal{B}(\mathbb{R}))$, and let $F=F_{\mu}$ be its distribution function.

(i) Monotonicity. For all $x,y\in\mathbb{R}$ with $x\le y$,

$$0\le F(x)\le F(y)\le1.$$

(ii) Right-continuity. For every $x\in\mathbb{R}$ and every $\varepsilon>0$ there is $\delta>0$ with

$$F(x)\le F(y)<F(x)+\varepsilon\qquad\text{for all }y\in\mathbb{R}\text{ with }x\le y<x+\delta.$$

(iii) Limit at $-\infty$. For every $\varepsilon>0$ there is $M>0$ with

$$F(x)<\varepsilon\qquad\text{for all }x\in\mathbb{R}\text{ with }x\le-M.$$

(iv) Limit at $\infty$. For every $\varepsilon>0$ there is $M>0$ with

$$F(x)>1-\varepsilon\qquad\text{for all }x\in\mathbb{R}\text{ with }x\ge M.$$

(v) The distribution function determines the measure. If $\nu$ is a probability measure on $(\mathbb{R},\mathcal{B}(\mathbb{R}))$ with $F_{\nu}=F_{\mu}$, then $\nu=\mu$.

(vi) Identically distributed random variables. Let $X$ be a random variable on $(\Omega,\mathcal{F},P)$ and $Y$ a random variable on a probability space $(\Omega',\mathcal{F}',P')$. Then $X$ and $Y$ are identically distributed if and only if $F_{X}=F_{Y}$.