Cumulative Distribution Function

The cumulative distribution function, or more simply the distribution function, $F$ of the random variable $X$ is defined for any real number $x$ by

$$F(x)=P(X\le x)=P(X^{-1}(-\infty ,x]).$$

Since $(-\infty ,x]$ is measurable, the above function is well-defined.

Properties

• (Monotonicity) $x_1\le x_2⟹F(x_1)\le F(x_2)$.
• (Limit) ${\lim }_{x\to -\infty }F(x)=0$, ${\lim }_{x\to \infty }F(x)=1$.
• (Right continuity) $F(x)=F(x^{+})$.

The proof of the right continuity uses the fact that given any decreasing sequence $\{x_n\}$ to $x$, we have ${\cap }_{n=1}^{\infty }\{X\le x_n\}=\{X\le x\}$. While given any increasing sequence $\{x_n\}$ to $x$, we have ${\cup }_{n=1}^{\infty }\{X\le x_n\}=\{X<x\}$. Therefore, a CDF needs not be left continuous, and

$$F(x)-F(x^{-})=P(X\le x)-P(X<x)=P(X=x).$$

Any function $F:\mathbb{R}\to [0,1]$ that satisfies the above properties is called a distribution function. The next section shows that any distribution function defines a Random Variable, or equivalently, its probability law.

• Moreover, the probability law and CDF (of a r.v.) uniquely determine each other. ^law-cdf
  • Proof sketch: law ⇒ CDF is trivial. Now suppose a CDF $F_X$ is given. It is easy to show that $F_X$ uniquely determines a probability measure $P_X$ on field ${\mathcal{F}}_0$, the collection of all unions of finitely many intervals of the form $(a,b]$. Moreover, one can show that $P_X$ satisfies the countable additivity property on ${\mathcal{F}}_0$. Then by Caratheodory’s Extension, $P_X$ can be uniquely extended to a probability measure on the $(\mathbb{R},\mathcal{B})$.

Simulate Any Distribution

To simulate an arbitrary distribution $F$, let $X\sim \mathcal{U}(0,1)$, then let $Y=F^{-1}(X):=\inf \{y:F(y)\ge X\}$. Note that $F^{-1}$ is left-continuous. Then we have

$$P(Y\le y)=P(F^{-1}(X)\le y)=P(X\le F(y))=F_{\mathcal{U}(0,1)}(F(y))=F(y),$$

where the second equality comes from the property of the general inverse $F(y)\ge x⟺y\ge F^{-1}(x)$ Therefore, $Y\sim F$.